# Whole Coefficient Laws, Independent Endpoints and Zero-Infinity-Prefix Radii

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(1/2)^n for every n≥0, positive ratio q=1/2, and center c=-12. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(1/2)^n_for_every_n>=0, -12, 2, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(1/2)^n_for_every_n>=0, -12, 2, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0001/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 2. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(2/3)^n for every n≥0, positive ratio q=2/3, and center c=-11. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(2/3)^n_for_every_n>=0, -11, 3/2, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(2/3)^n_for_every_n>=0, -11, 3/2, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0002/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 3. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(3/4)^n for every n≥0, positive ratio q=3/4, and center c=-10. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(3/4)^n_for_every_n>=0, -10, 4/3, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(3/4)^n_for_every_n>=0, -10, 4/3, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0003/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 4. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(4/5)^n for every n≥0, positive ratio q=4/5, and center c=-9. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(4/5)^n_for_every_n>=0, -9, 5/4, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(4/5)^n_for_every_n>=0, -9, 5/4, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0004/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 5. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(5/6)^n for every n≥0, positive ratio q=5/6, and center c=-8. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(5/6)^n_for_every_n>=0, -8, 6/5, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(5/6)^n_for_every_n>=0, -8, 6/5, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0005/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 6. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(6/7)^n for every n≥0, positive ratio q=6/7, and center c=-7. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(6/7)^n_for_every_n>=0, -7, 7/6, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(6/7)^n_for_every_n>=0, -7, 7/6, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0006/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 7. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(7/8)^n for every n≥0, positive ratio q=7/8, and center c=-6. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(7/8)^n_for_every_n>=0, -6, 8/7, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(7/8)^n_for_every_n>=0, -6, 8/7, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0007/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 8. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(8/9)^n for every n≥0, positive ratio q=8/9, and center c=-5. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(8/9)^n_for_every_n>=0, -5, 9/8, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(8/9)^n_for_every_n>=0, -5, 9/8, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0008/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 9. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(9/10)^n for every n≥0, positive ratio q=9/10, and center c=-4. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(9/10)^n_for_every_n>=0, -4, 10/9, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(9/10)^n_for_every_n>=0, -4, 10/9, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0009/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 10. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(10/11)^n for every n≥0, positive ratio q=10/11, and center c=-3. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(10/11)^n_for_every_n>=0, -3, 11/10, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(10/11)^n_for_every_n>=0, -3, 11/10, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0010/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 11. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(11/12)^n for every n≥0, positive ratio q=11/12, and center c=-2. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(11/12)^n_for_every_n>=0, -2, 12/11, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(11/12)^n_for_every_n>=0, -2, 12/11, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0011/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 12. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(12/13)^n for every n≥0, positive ratio q=12/13, and center c=-1. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(12/13)^n_for_every_n>=0, -1, 13/12, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(12/13)^n_for_every_n>=0, -1, 13/12, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0012/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 13. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(13/14)^n for every n≥0, positive ratio q=13/14, and center c=0. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(13/14)^n_for_every_n>=0, 0, 14/13, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(13/14)^n_for_every_n>=0, 0, 14/13, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0013/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 14. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(14/15)^n for every n≥0, positive ratio q=14/15, and center c=1. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(14/15)^n_for_every_n>=0, 1, 15/14, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(14/15)^n_for_every_n>=0, 1, 15/14, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0014/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 15. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(15/16)^n for every n≥0, positive ratio q=15/16, and center c=2. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(15/16)^n_for_every_n>=0, 2, 16/15, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(15/16)^n_for_every_n>=0, 2, 16/15, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0015/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 16. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(16/17)^n for every n≥0, positive ratio q=16/17, and center c=3. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(16/17)^n_for_every_n>=0, 3, 17/16, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(16/17)^n_for_every_n>=0, 3, 17/16, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0016/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 17. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(17/18)^n for every n≥0, positive ratio q=17/18, and center c=4. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(17/18)^n_for_every_n>=0, 4, 18/17, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(17/18)^n_for_every_n>=0, 4, 18/17, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0017/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 18. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(18/19)^n for every n≥0, positive ratio q=18/19, and center c=5. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(18/19)^n_for_every_n>=0, 5, 19/18, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(18/19)^n_for_every_n>=0, 5, 19/18, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0018/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 19. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(19/20)^n for every n≥0, positive ratio q=19/20, and center c=6. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(19/20)^n_for_every_n>=0, 6, 20/19, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(19/20)^n_for_every_n>=0, 6, 20/19, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0019/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 20. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(20/21)^n for every n≥0, positive ratio q=20/21, and center c=7. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(20/21)^n_for_every_n>=0, 7, 21/20, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(20/21)^n_for_every_n>=0, 7, 21/20, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0020/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 21. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(21/22)^n for every n≥0, positive ratio q=21/22, and center c=8. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(21/22)^n_for_every_n>=0, 8, 22/21, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(21/22)^n_for_every_n>=0, 8, 22/21, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0021/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 22. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(22/23)^n for every n≥0, positive ratio q=22/23, and center c=9. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(22/23)^n_for_every_n>=0, 9, 23/22, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(22/23)^n_for_every_n>=0, 9, 23/22, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0022/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 23. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(23/24)^n for every n≥0, positive ratio q=23/24, and center c=10. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(23/24)^n_for_every_n>=0, 10, 24/23, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(23/24)^n_for_every_n>=0, 10, 24/23, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0023/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 24. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(24/25)^n for every n≥0, positive ratio q=24/25, and center c=11. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(24/25)^n_for_every_n>=0, 11, 25/24, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(24/25)^n_for_every_n>=0, 11, 25/24, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0024/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 25. A Whole Geometric Coefficient Law Gives an Exact Radius

Use a_n=(25/26)^n for every n≥0, positive ratio q=25/26, and center c=12. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).

View answer and explanation(a_n=(25/26)^n_for_every_n>=0, 12, 26/25, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(25/26)^n_for_every_n>=0, 12, 26/25, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0025/)

Radius of convergence of a power series · qf-mod43-radius-geometric-exact · candidate





## 26. Classify the Exact Distance from the Declared Center

Use a_n=(1/2)^n for every n≥0, positive q=1/2, center=-12, point=-12. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(1/2)^n_for_every_n>=0, -12, -12, 0, 2, absolute_convergence, Yes, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(1/2)^n_for_every_n>=0, -12, -12, 0, 2, absolute_convergence, Yes, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0026/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 27. Classify the Exact Distance from the Declared Center

Use a_n=(2/3)^n for every n≥0, positive q=2/3, center=-11, point=-9. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(2/3)^n_for_every_n>=0, -11, -9, 2, 3/2, divergence, No, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(2/3)^n_for_every_n>=0, -11, -9, 2, 3/2, divergence, No, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0027/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 28. Classify the Exact Distance from the Declared Center

Use a_n=(3/4)^n for every n≥0, positive q=3/4, center=-10, point=-10. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(3/4)^n_for_every_n>=0, -10, -10, 0, 4/3, absolute_convergence, Yes, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(3/4)^n_for_every_n>=0, -10, -10, 0, 4/3, absolute_convergence, Yes, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0028/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 29. Classify the Exact Distance from the Declared Center

Use a_n=(4/5)^n for every n≥0, positive q=4/5, center=-9, point=-7. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(4/5)^n_for_every_n>=0, -9, -7, 2, 5/4, divergence, No, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(4/5)^n_for_every_n>=0, -9, -7, 2, 5/4, divergence, No, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0029/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 30. Classify the Exact Distance from the Declared Center

Use a_n=(5/6)^n for every n≥0, positive q=5/6, center=-8, point=-8. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(5/6)^n_for_every_n>=0, -8, -8, 0, 6/5, absolute_convergence, Yes, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(5/6)^n_for_every_n>=0, -8, -8, 0, 6/5, absolute_convergence, Yes, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0030/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 31. Classify the Exact Distance from the Declared Center

Use a_n=(6/7)^n for every n≥0, positive q=6/7, center=-7, point=-5. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(6/7)^n_for_every_n>=0, -7, -5, 2, 7/6, divergence, No, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(6/7)^n_for_every_n>=0, -7, -5, 2, 7/6, divergence, No, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0031/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 32. Classify the Exact Distance from the Declared Center

Use a_n=(7/8)^n for every n≥0, positive q=7/8, center=-6, point=-6. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(7/8)^n_for_every_n>=0, -6, -6, 0, 8/7, absolute_convergence, Yes, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(7/8)^n_for_every_n>=0, -6, -6, 0, 8/7, absolute_convergence, Yes, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0032/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 33. Classify the Exact Distance from the Declared Center

Use a_n=(8/9)^n for every n≥0, positive q=8/9, center=-5, point=-3. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(8/9)^n_for_every_n>=0, -5, -3, 2, 9/8, divergence, No, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(8/9)^n_for_every_n>=0, -5, -3, 2, 9/8, divergence, No, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0033/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 34. Classify the Exact Distance from the Declared Center

Use a_n=(9/10)^n for every n≥0, positive q=9/10, center=-4, point=-4. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(9/10)^n_for_every_n>=0, -4, -4, 0, 10/9, absolute_convergence, Yes, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(9/10)^n_for_every_n>=0, -4, -4, 0, 10/9, absolute_convergence, Yes, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0034/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 35. Classify the Exact Distance from the Declared Center

Use a_n=(10/11)^n for every n≥0, positive q=10/11, center=-3, point=-1. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(10/11)^n_for_every_n>=0, -3, -1, 2, 11/10, divergence, No, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(10/11)^n_for_every_n>=0, -3, -1, 2, 11/10, divergence, No, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0035/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 36. Classify the Exact Distance from the Declared Center

Use a_n=(11/12)^n for every n≥0, positive q=11/12, center=-2, point=-2. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(11/12)^n_for_every_n>=0, -2, -2, 0, 12/11, absolute_convergence, Yes, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(11/12)^n_for_every_n>=0, -2, -2, 0, 12/11, absolute_convergence, Yes, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0036/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 37. Classify the Exact Distance from the Declared Center

Use a_n=(12/13)^n for every n≥0, positive q=12/13, center=-1, point=1. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(12/13)^n_for_every_n>=0, -1, 1, 2, 13/12, divergence, No, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(12/13)^n_for_every_n>=0, -1, 1, 2, 13/12, divergence, No, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0037/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 38. Classify the Exact Distance from the Declared Center

Use a_n=(13/14)^n for every n≥0, positive q=13/14, center=0, point=0. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(13/14)^n_for_every_n>=0, 0, 0, 0, 14/13, absolute_convergence, Yes, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(13/14)^n_for_every_n>=0, 0, 0, 0, 14/13, absolute_convergence, Yes, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0038/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 39. Classify the Exact Distance from the Declared Center

Use a_n=(14/15)^n for every n≥0, positive q=14/15, center=1, point=3. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(14/15)^n_for_every_n>=0, 1, 3, 2, 15/14, divergence, No, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(14/15)^n_for_every_n>=0, 1, 3, 2, 15/14, divergence, No, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0039/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 40. Classify the Exact Distance from the Declared Center

Use a_n=(15/16)^n for every n≥0, positive q=15/16, center=2, point=2. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(15/16)^n_for_every_n>=0, 2, 2, 0, 16/15, absolute_convergence, Yes, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(15/16)^n_for_every_n>=0, 2, 2, 0, 16/15, absolute_convergence, Yes, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0040/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 41. Classify the Exact Distance from the Declared Center

Use a_n=(16/17)^n for every n≥0, positive q=16/17, center=3, point=5. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(16/17)^n_for_every_n>=0, 3, 5, 2, 17/16, divergence, No, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(16/17)^n_for_every_n>=0, 3, 5, 2, 17/16, divergence, No, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0041/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 42. Classify the Exact Distance from the Declared Center

Use a_n=(17/18)^n for every n≥0, positive q=17/18, center=4, point=4. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(17/18)^n_for_every_n>=0, 4, 4, 0, 18/17, absolute_convergence, Yes, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(17/18)^n_for_every_n>=0, 4, 4, 0, 18/17, absolute_convergence, Yes, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0042/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 43. Classify the Exact Distance from the Declared Center

Use a_n=(18/19)^n for every n≥0, positive q=18/19, center=5, point=7. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(18/19)^n_for_every_n>=0, 5, 7, 2, 19/18, divergence, No, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(18/19)^n_for_every_n>=0, 5, 7, 2, 19/18, divergence, No, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0043/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 44. Classify the Exact Distance from the Declared Center

Use a_n=(19/20)^n for every n≥0, positive q=19/20, center=6, point=6. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(19/20)^n_for_every_n>=0, 6, 6, 0, 20/19, absolute_convergence, Yes, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(19/20)^n_for_every_n>=0, 6, 6, 0, 20/19, absolute_convergence, Yes, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0044/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 45. Classify the Exact Distance from the Declared Center

Use a_n=(20/21)^n for every n≥0, positive q=20/21, center=7, point=9. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(20/21)^n_for_every_n>=0, 7, 9, 2, 21/20, divergence, No, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(20/21)^n_for_every_n>=0, 7, 9, 2, 21/20, divergence, No, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0045/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 46. Classify the Exact Distance from the Declared Center

Use a_n=(21/22)^n for every n≥0, positive q=21/22, center=8, point=8. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(21/22)^n_for_every_n>=0, 8, 8, 0, 22/21, absolute_convergence, Yes, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(21/22)^n_for_every_n>=0, 8, 8, 0, 22/21, absolute_convergence, Yes, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0046/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 47. Classify the Exact Distance from the Declared Center

Use a_n=(22/23)^n for every n≥0, positive q=22/23, center=9, point=11. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(22/23)^n_for_every_n>=0, 9, 11, 2, 23/22, divergence, No, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(22/23)^n_for_every_n>=0, 9, 11, 2, 23/22, divergence, No, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0047/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 48. Classify the Exact Distance from the Declared Center

Use a_n=(23/24)^n for every n≥0, positive q=23/24, center=10, point=10. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(23/24)^n_for_every_n>=0, 10, 10, 0, 24/23, absolute_convergence, Yes, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(23/24)^n_for_every_n>=0, 10, 10, 0, 24/23, absolute_convergence, Yes, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0048/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 49. Classify the Exact Distance from the Declared Center

Use a_n=(24/25)^n for every n≥0, positive q=24/25, center=11, point=13. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(24/25)^n_for_every_n>=0, 11, 13, 2, 25/24, divergence, No, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(24/25)^n_for_every_n>=0, 11, 13, 2, 25/24, divergence, No, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0049/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 50. Classify the Exact Distance from the Declared Center

Use a_n=(25/26)^n for every n≥0, positive q=25/26, center=12, point=12. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

View answer and explanation(a_n=(25/26)^n_for_every_n>=0, 12, 12, 0, 26/25, absolute_convergence, Yes, classification_uses_exact_distance_from_center)

Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(25/26)^n_for_every_n>=0, 12, 12, 0, 26/25, absolute_convergence, Yes, classification_uses_exact_distance_from_center).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0050/)

Radius of convergence of a power series · qf-mod43-radius-inside-outside-audit · candidate





## 51. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=1^(−n)/(n+1)^0 for every n≥0 and center=-12, with radius R=1>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=1^(-n)/(n+1)^0_for_every_n>=0, -12, 1, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=1^(-n)/(n+1)^0_for_every_n>=0, -12, 1, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0051/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 52. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=2^(−n)/(n+1)^1 for every n≥0 and center=-11, with radius R=2>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=2^(-n)/(n+1)^1_for_every_n>=0, -11, 2, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=2^(-n)/(n+1)^1_for_every_n>=0, -11, 2, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0052/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 53. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=3^(−n)/(n+1)^2 for every n≥0 and center=-10, with radius R=3>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=3^(-n)/(n+1)^2_for_every_n>=0, -10, 3, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=3^(-n)/(n+1)^2_for_every_n>=0, -10, 3, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0053/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 54. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=4^(−n)/(n+1)^0 for every n≥0 and center=-9, with radius R=4>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=4^(-n)/(n+1)^0_for_every_n>=0, -9, 4, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=4^(-n)/(n+1)^0_for_every_n>=0, -9, 4, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0054/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 55. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=5^(−n)/(n+1)^1 for every n≥0 and center=-8, with radius R=5>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=5^(-n)/(n+1)^1_for_every_n>=0, -8, 5, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=5^(-n)/(n+1)^1_for_every_n>=0, -8, 5, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0055/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 56. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=1^(−n)/(n+1)^2 for every n≥0 and center=-7, with radius R=1>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=1^(-n)/(n+1)^2_for_every_n>=0, -7, 1, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=1^(-n)/(n+1)^2_for_every_n>=0, -7, 1, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0056/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 57. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=2^(−n)/(n+1)^0 for every n≥0 and center=-6, with radius R=2>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=2^(-n)/(n+1)^0_for_every_n>=0, -6, 2, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=2^(-n)/(n+1)^0_for_every_n>=0, -6, 2, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0057/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 58. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=3^(−n)/(n+1)^1 for every n≥0 and center=-5, with radius R=3>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=3^(-n)/(n+1)^1_for_every_n>=0, -5, 3, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=3^(-n)/(n+1)^1_for_every_n>=0, -5, 3, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0058/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 59. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=4^(−n)/(n+1)^2 for every n≥0 and center=-4, with radius R=4>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=4^(-n)/(n+1)^2_for_every_n>=0, -4, 4, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=4^(-n)/(n+1)^2_for_every_n>=0, -4, 4, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0059/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 60. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=5^(−n)/(n+1)^0 for every n≥0 and center=-3, with radius R=5>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=5^(-n)/(n+1)^0_for_every_n>=0, -3, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=5^(-n)/(n+1)^0_for_every_n>=0, -3, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0060/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 61. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=1^(−n)/(n+1)^1 for every n≥0 and center=-2, with radius R=1>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=1^(-n)/(n+1)^1_for_every_n>=0, -2, 1, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=1^(-n)/(n+1)^1_for_every_n>=0, -2, 1, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0061/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 62. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=2^(−n)/(n+1)^2 for every n≥0 and center=-1, with radius R=2>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=2^(-n)/(n+1)^2_for_every_n>=0, -1, 2, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=2^(-n)/(n+1)^2_for_every_n>=0, -1, 2, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0062/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 63. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=3^(−n)/(n+1)^0 for every n≥0 and center=0, with radius R=3>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=3^(-n)/(n+1)^0_for_every_n>=0, 0, 3, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=3^(-n)/(n+1)^0_for_every_n>=0, 0, 3, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0063/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 64. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=4^(−n)/(n+1)^1 for every n≥0 and center=1, with radius R=4>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=4^(-n)/(n+1)^1_for_every_n>=0, 1, 4, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=4^(-n)/(n+1)^1_for_every_n>=0, 1, 4, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0064/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 65. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=5^(−n)/(n+1)^2 for every n≥0 and center=2, with radius R=5>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=5^(-n)/(n+1)^2_for_every_n>=0, 2, 5, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=5^(-n)/(n+1)^2_for_every_n>=0, 2, 5, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0065/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 66. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=1^(−n)/(n+1)^0 for every n≥0 and center=3, with radius R=1>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=1^(-n)/(n+1)^0_for_every_n>=0, 3, 1, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=1^(-n)/(n+1)^0_for_every_n>=0, 3, 1, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0066/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 67. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=2^(−n)/(n+1)^1 for every n≥0 and center=4, with radius R=2>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=2^(-n)/(n+1)^1_for_every_n>=0, 4, 2, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=2^(-n)/(n+1)^1_for_every_n>=0, 4, 2, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0067/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 68. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=3^(−n)/(n+1)^2 for every n≥0 and center=5, with radius R=3>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=3^(-n)/(n+1)^2_for_every_n>=0, 5, 3, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=3^(-n)/(n+1)^2_for_every_n>=0, 5, 3, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0068/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 69. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=4^(−n)/(n+1)^0 for every n≥0 and center=6, with radius R=4>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=4^(-n)/(n+1)^0_for_every_n>=0, 6, 4, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=4^(-n)/(n+1)^0_for_every_n>=0, 6, 4, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0069/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 70. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=5^(−n)/(n+1)^1 for every n≥0 and center=7, with radius R=5>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=5^(-n)/(n+1)^1_for_every_n>=0, 7, 5, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=5^(-n)/(n+1)^1_for_every_n>=0, 7, 5, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0070/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 71. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=1^(−n)/(n+1)^2 for every n≥0 and center=8, with radius R=1>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=1^(-n)/(n+1)^2_for_every_n>=0, 8, 1, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=1^(-n)/(n+1)^2_for_every_n>=0, 8, 1, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0071/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 72. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=2^(−n)/(n+1)^0 for every n≥0 and center=9, with radius R=2>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=2^(-n)/(n+1)^0_for_every_n>=0, 9, 2, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=2^(-n)/(n+1)^0_for_every_n>=0, 9, 2, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0072/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 73. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=3^(−n)/(n+1)^1 for every n≥0 and center=10, with radius R=3>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=3^(-n)/(n+1)^1_for_every_n>=0, 10, 3, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=3^(-n)/(n+1)^1_for_every_n>=0, 10, 3, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0073/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 74. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=4^(−n)/(n+1)^2 for every n≥0 and center=11, with radius R=4>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=4^(-n)/(n+1)^2_for_every_n>=0, 11, 4, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=4^(-n)/(n+1)^2_for_every_n>=0, 11, 4, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0074/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 75. The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Use a_n=5^(−n)/(n+1)^0 for every n≥0 and center=12, with radius R=5>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).

positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification.

View answer and explanation(a_n=5^(-n)/(n+1)^0_for_every_n>=0, 12, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)

At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=5^(-n)/(n+1)^0_for_every_n>=0, 12, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0075/)

Radius of convergence of a power series · qf-mod43-radius-endpoint-independent · candidate





## 76. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=zero_radius, positive scale=1, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(zero_radius, 1, a_n=1*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 1, a_n=1*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0076/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 77. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=infinite_radius, positive scale=2, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(infinite_radius, 2, a_n=2/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 2, a_n=2/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0077/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 78. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=finite_prefix, positive scale=3, source prefix_length=1, first_ratio=1/2, second_ratio=1/3. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(finite_prefix, 3, 1, [0, 0], a_n=0_for_n<=1;a_n=3*(1/2)^(n-2)_for_n>1, b_n=0_for_n<=1;b_n=3*(1/3)^(n-2)_for_n>1, 2, 3, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 3, 1, [0, 0], a_n=0_for_n<=1;a_n=3*(1/2)^(n-2)_for_n>1, b_n=0_for_n<=1;b_n=3*(1/3)^(n-2)_for_n>1, 2, 3, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0078/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 79. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=zero_radius, positive scale=4, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(zero_radius, 4, a_n=4*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 4, a_n=4*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0079/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 80. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=infinite_radius, positive scale=5, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(infinite_radius, 5, a_n=5/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 5, a_n=5/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0080/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 81. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=finite_prefix, positive scale=6, source prefix_length=2, first_ratio=1/2, second_ratio=1/4. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(finite_prefix, 6, 2, [0, 0, 0], a_n=0_for_n<=2;a_n=6*(1/2)^(n-3)_for_n>2, b_n=0_for_n<=2;b_n=6*(1/4)^(n-3)_for_n>2, 2, 4, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 6, 2, [0, 0, 0], a_n=0_for_n<=2;a_n=6*(1/2)^(n-3)_for_n>2, b_n=0_for_n<=2;b_n=6*(1/4)^(n-3)_for_n>2, 2, 4, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0081/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 82. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=zero_radius, positive scale=7, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(zero_radius, 7, a_n=7*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 7, a_n=7*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0082/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 83. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=infinite_radius, positive scale=8, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(infinite_radius, 8, a_n=8/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 8, a_n=8/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0083/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 84. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=finite_prefix, positive scale=9, source prefix_length=3, first_ratio=1/2, second_ratio=1/5. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(finite_prefix, 9, 3, [0, 0, 0, 0], a_n=0_for_n<=3;a_n=9*(1/2)^(n-4)_for_n>3, b_n=0_for_n<=3;b_n=9*(1/5)^(n-4)_for_n>3, 2, 5, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 9, 3, [0, 0, 0, 0], a_n=0_for_n<=3;a_n=9*(1/2)^(n-4)_for_n>3, b_n=0_for_n<=3;b_n=9*(1/5)^(n-4)_for_n>3, 2, 5, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0084/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 85. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=zero_radius, positive scale=10, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(zero_radius, 10, a_n=10*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 10, a_n=10*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0085/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 86. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=infinite_radius, positive scale=11, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(infinite_radius, 11, a_n=11/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 11, a_n=11/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0086/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 87. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=finite_prefix, positive scale=12, source prefix_length=4, first_ratio=1/2, second_ratio=1/6. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(finite_prefix, 12, 4, [0, 0, 0, 0, 0], a_n=0_for_n<=4;a_n=12*(1/2)^(n-5)_for_n>4, b_n=0_for_n<=4;b_n=12*(1/6)^(n-5)_for_n>4, 2, 6, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 12, 4, [0, 0, 0, 0, 0], a_n=0_for_n<=4;a_n=12*(1/2)^(n-5)_for_n>4, b_n=0_for_n<=4;b_n=12*(1/6)^(n-5)_for_n>4, 2, 6, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0087/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 88. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=zero_radius, positive scale=13, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(zero_radius, 13, a_n=13*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 13, a_n=13*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0088/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 89. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=infinite_radius, positive scale=14, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(infinite_radius, 14, a_n=14/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 14, a_n=14/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0089/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 90. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=finite_prefix, positive scale=15, source prefix_length=5, first_ratio=1/2, second_ratio=1/7. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(finite_prefix, 15, 5, [0, 0, 0, 0, 0, 0], a_n=0_for_n<=5;a_n=15*(1/2)^(n-6)_for_n>5, b_n=0_for_n<=5;b_n=15*(1/7)^(n-6)_for_n>5, 2, 7, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 15, 5, [0, 0, 0, 0, 0, 0], a_n=0_for_n<=5;a_n=15*(1/2)^(n-6)_for_n>5, b_n=0_for_n<=5;b_n=15*(1/7)^(n-6)_for_n>5, 2, 7, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0090/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 91. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=zero_radius, positive scale=16, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(zero_radius, 16, a_n=16*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 16, a_n=16*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0091/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 92. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=infinite_radius, positive scale=17, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(infinite_radius, 17, a_n=17/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 17, a_n=17/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0092/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 93. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=finite_prefix, positive scale=18, source prefix_length=6, first_ratio=1/2, second_ratio=1/8. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(finite_prefix, 18, 6, [0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=6;a_n=18*(1/2)^(n-7)_for_n>6, b_n=0_for_n<=6;b_n=18*(1/8)^(n-7)_for_n>6, 2, 8, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 18, 6, [0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=6;a_n=18*(1/2)^(n-7)_for_n>6, b_n=0_for_n<=6;b_n=18*(1/8)^(n-7)_for_n>6, 2, 8, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0093/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 94. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=zero_radius, positive scale=19, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(zero_radius, 19, a_n=19*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 19, a_n=19*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0094/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 95. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=infinite_radius, positive scale=20, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(infinite_radius, 20, a_n=20/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 20, a_n=20/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0095/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 96. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=finite_prefix, positive scale=21, source prefix_length=7, first_ratio=1/2, second_ratio=1/9. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(finite_prefix, 21, 7, [0, 0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=7;a_n=21*(1/2)^(n-8)_for_n>7, b_n=0_for_n<=7;b_n=21*(1/9)^(n-8)_for_n>7, 2, 9, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 21, 7, [0, 0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=7;a_n=21*(1/2)^(n-8)_for_n>7, b_n=0_for_n<=7;b_n=21*(1/9)^(n-8)_for_n>7, 2, 9, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0096/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 97. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=zero_radius, positive scale=22, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(zero_radius, 22, a_n=22*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 22, a_n=22*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0097/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 98. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=infinite_radius, positive scale=23, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(infinite_radius, 23, a_n=23/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 23, a_n=23/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0098/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 99. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=finite_prefix, positive scale=24, source prefix_length=8, first_ratio=1/2, second_ratio=1/10. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(finite_prefix, 24, 8, [0, 0, 0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=8;a_n=24*(1/2)^(n-9)_for_n>8, b_n=0_for_n<=8;b_n=24*(1/10)^(n-9)_for_n>8, 2, 10, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 24, 8, [0, 0, 0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=8;a_n=24*(1/2)^(n-9)_for_n>8, b_n=0_for_n<=8;b_n=24*(1/10)^(n-9)_for_n>8, 2, 10, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0099/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 100. Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Use mode=zero_radius, positive scale=25, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii.

View answer and explanation(zero_radius, 25, a_n=25*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)

zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 25, a_n=25*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0100/)

Radius of convergence of a power series · qf-mod43-radius-zero-infinity-prefix-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations/validation-report.json)