# Nonnegative Tail Comparisons and Correct Convergence Directions

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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-20261015-advance-20261117-r01-series-comparison-test-foundations · 2026-10-15 · 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 Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=1,B=2, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(1, 2, 1/7, 7/6, 7/3, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (1, 2, 1/7, 7/6, 7/3, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0001/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 2. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=2,B=3, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(2, 3, 1/4, 8/3, 4, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (2, 3, 1/4, 8/3, 4, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0002/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 3. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=3,B=4, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(3, 4, 1/3, 9/2, 6, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (3, 4, 1/3, 9/2, 6, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0003/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 4. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=4,B=5, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(4, 5, 2/5, 20/3, 25/3, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (4, 5, 2/5, 20/3, 25/3, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0004/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 5. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=5,B=6, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(5, 6, 5/11, 55/6, 11, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (5, 6, 5/11, 55/6, 11, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0005/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 6. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=6,B=7, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(6, 7, 1/7, 7, 49/6, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (6, 7, 1/7, 7, 49/6, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0006/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 7. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=7,B=8, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(7, 8, 1/4, 28/3, 32/3, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (7, 8, 1/4, 28/3, 32/3, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0007/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 8. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=8,B=9, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(8, 9, 1/3, 12, 27/2, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (8, 9, 1/3, 12, 27/2, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0008/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 9. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=9,B=10, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(9, 10, 2/5, 15, 50/3, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (9, 10, 2/5, 15, 50/3, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0009/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 10. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=10,B=11, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(10, 11, 5/11, 55/3, 121/6, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (10, 11, 5/11, 55/3, 121/6, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0010/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 11. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=11,B=12, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(11, 12, 1/7, 77/6, 14, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (11, 12, 1/7, 77/6, 14, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0011/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 12. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=12,B=13, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(12, 13, 1/4, 16, 52/3, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (12, 13, 1/4, 16, 52/3, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0012/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 13. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=13,B=14, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(13, 14, 1/3, 39/2, 21, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (13, 14, 1/3, 39/2, 21, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0013/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 14. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=14,B=15, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(14, 15, 2/5, 70/3, 25, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (14, 15, 2/5, 70/3, 25, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0014/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 15. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=15,B=16, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(15, 16, 5/11, 55/2, 88/3, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (15, 16, 5/11, 55/2, 88/3, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0015/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 16. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=16,B=17, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(16, 17, 1/7, 56/3, 119/6, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (16, 17, 1/7, 56/3, 119/6, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0016/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 17. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=17,B=18, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(17, 18, 1/4, 68/3, 24, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (17, 18, 1/4, 68/3, 24, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0017/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 18. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=18,B=19, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(18, 19, 1/3, 27, 57/2, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (18, 19, 1/3, 27, 57/2, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0018/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 19. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=19,B=20, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(19, 20, 2/5, 95/3, 100/3, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (19, 20, 2/5, 95/3, 100/3, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0019/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 20. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=20,B=21, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(20, 21, 5/11, 110/3, 77/2, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (20, 21, 5/11, 110/3, 77/2, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0020/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 21. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=21,B=22, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(21, 22, 1/7, 49/2, 77/3, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (21, 22, 1/7, 49/2, 77/3, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0021/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 22. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=22,B=23, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(22, 23, 1/4, 88/3, 92/3, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (22, 23, 1/4, 88/3, 92/3, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0022/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 23. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=23,B=24, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(23, 24, 1/3, 69/2, 36, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (23, 24, 1/3, 69/2, 36, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0023/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 24. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=24,B=25, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(24, 25, 2/5, 40, 125/3, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (24, 25, 2/5, 40, 125/3, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0024/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 25. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

Use exact scales A=25,B=26, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

View answer and explanation(25, 26, 5/11, 275/6, 143/3, Yes, upper_comparison_proves_convergence)

Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (25, 26, 5/11, 275/6, 143/3, Yes, upper_comparison_proves_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0025/)

Comparison tests for series · qf-mod48-series-upper-geometric · candidate





## 26. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=2,B=1. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(2, 1, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (2, 1, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0026/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 27. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=4,B=2. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(4, 2, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (4, 2, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0027/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 28. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=6,B=3. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(6, 3, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (6, 3, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0028/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 29. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=8,B=4. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(8, 4, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (8, 4, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0029/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 30. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=10,B=5. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(10, 5, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (10, 5, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0030/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 31. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=12,B=6. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(12, 6, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (12, 6, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0031/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 32. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=14,B=7. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(14, 7, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (14, 7, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0032/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 33. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=16,B=8. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(16, 8, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (16, 8, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0033/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 34. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=18,B=9. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(18, 9, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (18, 9, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0034/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 35. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=20,B=10. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(20, 10, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (20, 10, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0035/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 36. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=22,B=11. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(22, 11, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (22, 11, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0036/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 37. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=24,B=12. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(24, 12, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (24, 12, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0037/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 38. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=26,B=13. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(26, 13, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (26, 13, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0038/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 39. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=28,B=14. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(28, 14, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (28, 14, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0039/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 40. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=30,B=15. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(30, 15, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (30, 15, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0040/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 41. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=32,B=16. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(32, 16, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (32, 16, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0041/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 42. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=34,B=17. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(34, 17, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (34, 17, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0042/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 43. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=36,B=18. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(36, 18, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (36, 18, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0043/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 44. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=38,B=19. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(38, 19, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (38, 19, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0044/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 45. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=40,B=20. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(40, 20, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (40, 20, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0045/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 46. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=42,B=21. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(42, 21, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (42, 21, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0046/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 47. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=44,B=22. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(44, 22, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (44, 22, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0047/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 48. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=46,B=23. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(46, 23, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (46, 23, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0048/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 49. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=48,B=24. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(48, 24, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (48, 24, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0049/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 50. A Positive Constant Lower Bound Gives Unbounded Partial Sums

Use exact scales A=50,B=25. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

View answer and explanation(50, 25, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (50, 25, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0050/)

Comparison tests for series · qf-mod48-series-lower-divergence · candidate





## 51. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=1,B=2, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(0, 0, 0, 8/7, 16/7, 8/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 8/7, 16/7, 8/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0051/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 52. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=2,B=3, rational ratio r=2/9. Prefix a=[2], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(1, 2, 1, 18/7, 27/7, 32/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 2, 1, 18/7, 27/7, 32/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0052/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 53. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=3,B=4, rational ratio r=3/10. Prefix a=[3, 4], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(2, 7, 3, 30/7, 40/7, 79/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 7, 3, 30/7, 40/7, 79/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0053/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 54. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=4,B=5, rational ratio r=4/11. Prefix a=[4, 5, 6], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(3, 15, 6, 44/7, 55/7, 149/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 15, 6, 44/7, 55/7, 149/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0054/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 55. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=5,B=6, rational ratio r=5/12. Prefix a=[5, 6, 7, 8], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(4, 26, 10, 60/7, 72/7, 242/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 26, 10, 60/7, 72/7, 242/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0055/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 56. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=6,B=7, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(0, 0, 0, 48/7, 8, 48/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 48/7, 8, 48/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0056/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 57. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=7,B=8, rational ratio r=2/9. Prefix a=[7], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(1, 7, 1, 9, 72/7, 16, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 7, 1, 9, 72/7, 16, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0057/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 58. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=8,B=9, rational ratio r=3/10. Prefix a=[8, 9], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(2, 17, 3, 80/7, 90/7, 199/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 17, 3, 80/7, 90/7, 199/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0058/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 59. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=9,B=10, rational ratio r=4/11. Prefix a=[9, 10, 11], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(3, 30, 6, 99/7, 110/7, 309/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 30, 6, 99/7, 110/7, 309/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0059/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 60. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=10,B=11, rational ratio r=5/12. Prefix a=[10, 11, 12, 13], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(4, 46, 10, 120/7, 132/7, 442/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 46, 10, 120/7, 132/7, 442/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0060/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 61. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=11,B=12, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(0, 0, 0, 88/7, 96/7, 88/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 88/7, 96/7, 88/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0061/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 62. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=12,B=13, rational ratio r=2/9. Prefix a=[12], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(1, 12, 1, 108/7, 117/7, 192/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 12, 1, 108/7, 117/7, 192/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0062/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 63. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=13,B=14, rational ratio r=3/10. Prefix a=[13, 14], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(2, 27, 3, 130/7, 20, 319/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 27, 3, 130/7, 20, 319/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0063/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 64. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=14,B=15, rational ratio r=4/11. Prefix a=[14, 15, 16], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(3, 45, 6, 22, 165/7, 67, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 45, 6, 22, 165/7, 67, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0064/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 65. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=15,B=16, rational ratio r=5/12. Prefix a=[15, 16, 17, 18], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(4, 66, 10, 180/7, 192/7, 642/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 66, 10, 180/7, 192/7, 642/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0065/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 66. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=16,B=17, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(0, 0, 0, 128/7, 136/7, 128/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 128/7, 136/7, 128/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0066/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 67. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=17,B=18, rational ratio r=2/9. Prefix a=[17], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(1, 17, 1, 153/7, 162/7, 272/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 17, 1, 153/7, 162/7, 272/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0067/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 68. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=18,B=19, rational ratio r=3/10. Prefix a=[18, 19], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(2, 37, 3, 180/7, 190/7, 439/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 37, 3, 180/7, 190/7, 439/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0068/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 69. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=19,B=20, rational ratio r=4/11. Prefix a=[19, 20, 21], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(3, 60, 6, 209/7, 220/7, 629/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 60, 6, 209/7, 220/7, 629/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0069/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 70. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=20,B=21, rational ratio r=5/12. Prefix a=[20, 21, 22, 23], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(4, 86, 10, 240/7, 36, 842/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 86, 10, 240/7, 36, 842/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0070/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 71. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=21,B=22, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(0, 0, 0, 24, 176/7, 24, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 24, 176/7, 24, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0071/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 72. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=22,B=23, rational ratio r=2/9. Prefix a=[22], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(1, 22, 1, 198/7, 207/7, 352/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 22, 1, 198/7, 207/7, 352/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0072/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 73. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=23,B=24, rational ratio r=3/10. Prefix a=[23, 24], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(2, 47, 3, 230/7, 240/7, 559/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 47, 3, 230/7, 240/7, 559/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0073/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 74. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=24,B=25, rational ratio r=4/11. Prefix a=[24, 25, 26], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(3, 75, 6, 264/7, 275/7, 789/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 75, 6, 264/7, 275/7, 789/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0074/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 75. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

Use exact scales A=25,B=26, rational ratio r=5/12. Prefix a=[25, 26, 27, 28], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.

View answer and explanation(4, 106, 10, 300/7, 312/7, 1042/7, Yes, finite_prefix_does_not_change_tail_convergence)

Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 106, 10, 300/7, 312/7, 1042/7, Yes, finite_prefix_does_not_change_tail_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0075/)

Comparison tests for series · qf-mod48-series-eventual-tail · candidate





## 76. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=finite_sample, positive scale=1. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(finite_sample, 1, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 1, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0076/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 77. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=divergent_upper, positive scale=2. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(divergent_upper, a_n=2/n^2, b_n=2/n, a_n<=b_n, b_diverges_but_a_converges, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=2/n^2, b_n=2/n, a_n<=b_n, b_diverges_but_a_converges, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0077/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 78. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=signed_terms, positive scale=3. Give (mode, series_description, failed_hypothesis, comparison_test_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(signed_terms, a_n=(-1)^n*3/n, nonnegative_hypothesis_fails, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*3/n, nonnegative_hypothesis_fails, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0078/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 79. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=unknown_reference, positive scale=4. Give (mode, scale, reference_status_scope, comparison_test_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(unknown_reference, 4, comparison_holds_reference_status_not_proved, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 4, comparison_holds_reference_status_not_proved, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0079/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 80. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=finite_sample, positive scale=5. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(finite_sample, 5, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 5, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0080/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 81. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=divergent_upper, positive scale=6. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(divergent_upper, a_n=6/n^2, b_n=6/n, a_n<=b_n, b_diverges_but_a_converges, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=6/n^2, b_n=6/n, a_n<=b_n, b_diverges_but_a_converges, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0081/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 82. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=signed_terms, positive scale=7. Give (mode, series_description, failed_hypothesis, comparison_test_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(signed_terms, a_n=(-1)^n*7/n, nonnegative_hypothesis_fails, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*7/n, nonnegative_hypothesis_fails, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0082/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 83. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=unknown_reference, positive scale=8. Give (mode, scale, reference_status_scope, comparison_test_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(unknown_reference, 8, comparison_holds_reference_status_not_proved, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 8, comparison_holds_reference_status_not_proved, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0083/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 84. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=finite_sample, positive scale=9. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(finite_sample, 9, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 9, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0084/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 85. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=divergent_upper, positive scale=10. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(divergent_upper, a_n=10/n^2, b_n=10/n, a_n<=b_n, b_diverges_but_a_converges, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=10/n^2, b_n=10/n, a_n<=b_n, b_diverges_but_a_converges, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0085/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 86. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=signed_terms, positive scale=11. Give (mode, series_description, failed_hypothesis, comparison_test_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(signed_terms, a_n=(-1)^n*11/n, nonnegative_hypothesis_fails, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*11/n, nonnegative_hypothesis_fails, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0086/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 87. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=unknown_reference, positive scale=12. Give (mode, scale, reference_status_scope, comparison_test_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(unknown_reference, 12, comparison_holds_reference_status_not_proved, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 12, comparison_holds_reference_status_not_proved, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0087/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 88. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=finite_sample, positive scale=13. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(finite_sample, 13, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 13, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0088/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 89. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=divergent_upper, positive scale=14. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(divergent_upper, a_n=14/n^2, b_n=14/n, a_n<=b_n, b_diverges_but_a_converges, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=14/n^2, b_n=14/n, a_n<=b_n, b_diverges_but_a_converges, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0089/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 90. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=signed_terms, positive scale=15. Give (mode, series_description, failed_hypothesis, comparison_test_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(signed_terms, a_n=(-1)^n*15/n, nonnegative_hypothesis_fails, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*15/n, nonnegative_hypothesis_fails, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0090/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 91. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=unknown_reference, positive scale=16. Give (mode, scale, reference_status_scope, comparison_test_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(unknown_reference, 16, comparison_holds_reference_status_not_proved, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 16, comparison_holds_reference_status_not_proved, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0091/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 92. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=finite_sample, positive scale=17. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(finite_sample, 17, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 17, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0092/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 93. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=divergent_upper, positive scale=18. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(divergent_upper, a_n=18/n^2, b_n=18/n, a_n<=b_n, b_diverges_but_a_converges, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=18/n^2, b_n=18/n, a_n<=b_n, b_diverges_but_a_converges, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0093/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 94. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=signed_terms, positive scale=19. Give (mode, series_description, failed_hypothesis, comparison_test_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(signed_terms, a_n=(-1)^n*19/n, nonnegative_hypothesis_fails, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*19/n, nonnegative_hypothesis_fails, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0094/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 95. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=unknown_reference, positive scale=20. Give (mode, scale, reference_status_scope, comparison_test_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(unknown_reference, 20, comparison_holds_reference_status_not_proved, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 20, comparison_holds_reference_status_not_proved, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0095/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 96. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=finite_sample, positive scale=21. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(finite_sample, 21, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 21, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0096/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 97. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=divergent_upper, positive scale=22. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(divergent_upper, a_n=22/n^2, b_n=22/n, a_n<=b_n, b_diverges_but_a_converges, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=22/n^2, b_n=22/n, a_n<=b_n, b_diverges_but_a_converges, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0097/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 98. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=signed_terms, positive scale=23. Give (mode, series_description, failed_hypothesis, comparison_test_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(signed_terms, a_n=(-1)^n*23/n, nonnegative_hypothesis_fails, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*23/n, nonnegative_hypothesis_fails, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0098/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 99. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=unknown_reference, positive scale=24. Give (mode, scale, reference_status_scope, comparison_test_claim).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(unknown_reference, 24, comparison_holds_reference_status_not_proved, No)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 24, comparison_holds_reference_status_not_proved, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0099/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate





## 100. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted

Boundary mode=finite_sample, positive scale=25. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).

The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.

View answer and explanation(finite_sample, 25, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)

finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 25, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r01-series-comparison-test-0100/)

Comparison tests for series · qf-mod48-series-comparison-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261117-r01-series-comparison-test-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261117-r01-series-comparison-test-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261117-r01-series-comparison-test-foundations/validation-report.json)