# Absolute Geometric Sums, Conditional Harmonic Tails and Finite Boundaries

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

batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations · 2026-10-14 · 100 questions

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

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

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



## 1. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=1/2, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(1/2, -1/2, 1, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (1/2, -1/2, 1, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0001/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 2. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=2/3, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(2/3, -1/2, 4/3, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (2/3, -1/2, 4/3, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0002/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 3. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=3/4, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(3/4, -1/2, 3/2, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (3/4, -1/2, 3/2, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0003/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 4. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=4/5, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(4/5, -1/2, 8/5, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (4/5, -1/2, 8/5, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0004/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 5. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=5/6, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(5/6, -1/2, 5/3, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (5/6, -1/2, 5/3, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0005/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 6. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=6/7, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(6/7, -1/2, 12/7, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (6/7, -1/2, 12/7, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0006/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 7. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=7/8, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(7/8, -1/2, 7/4, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (7/8, -1/2, 7/4, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0007/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 8. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=8/9, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(8/9, -1/2, 16/9, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (8/9, -1/2, 16/9, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0008/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 9. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=9/10, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(9/10, -1/2, 9/5, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (9/10, -1/2, 9/5, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0009/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 10. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=10/11, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(10/11, -1/2, 20/11, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (10/11, -1/2, 20/11, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0010/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 11. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=11/12, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(11/12, -1/2, 11/6, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (11/12, -1/2, 11/6, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0011/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 12. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=12/13, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(12/13, -1/2, 24/13, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (12/13, -1/2, 24/13, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0012/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 13. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=13/14, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(13/14, -1/2, 13/7, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (13/14, -1/2, 13/7, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0013/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 14. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=14/15, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(14/15, -1/2, 28/15, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (14/15, -1/2, 28/15, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0014/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 15. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=15/16, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(15/16, -1/2, 15/8, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (15/16, -1/2, 15/8, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0015/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 16. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=16/17, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(16/17, -1/2, 32/17, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (16/17, -1/2, 32/17, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0016/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 17. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=17/18, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(17/18, -1/2, 17/9, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (17/18, -1/2, 17/9, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0017/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 18. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=18/19, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(18/19, -1/2, 36/19, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (18/19, -1/2, 36/19, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0018/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 19. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=19/20, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(19/20, -1/2, 19/10, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (19/20, -1/2, 19/10, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0019/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 20. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=20/21, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(20/21, -1/2, 40/21, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (20/21, -1/2, 40/21, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0020/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 21. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=21/22, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(21/22, -1/2, 21/11, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (21/22, -1/2, 21/11, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0021/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 22. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=22/23, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(22/23, -1/2, 44/23, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (22/23, -1/2, 44/23, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0022/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 23. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=23/24, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(23/24, -1/2, 23/12, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (23/24, -1/2, 23/12, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0023/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 24. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=24/25, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(24/25, -1/2, 48/25, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (24/25, -1/2, 48/25, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0024/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 25. Sum Absolute Terms, Not the Absolute Value of the Signed Sum

Use terms a*r^k for k=0,1,... with exact a=25/26, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).

View answer and explanation(25/26, -1/2, 25/13, Yes, absolute_geometric_certificate)

The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (25/26, -1/2, 25/13, Yes, absolute_geometric_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0025/)

Absolute convergence of series · qf-mod35-absseries-geometric · candidate





## 26. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=1, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(1, 5, [1, -1/2, 1/3, -1/4, 1/5], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (1, 5, [1, -1/2, 1/3, -1/4, 1/5], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0026/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 27. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=2, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(2, 5, [1/2, -1/3, 1/4, -1/5, 1/6], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (2, 5, [1/2, -1/3, 1/4, -1/5, 1/6], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0027/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 28. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=3, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(3, 5, [1/3, -1/4, 1/5, -1/6, 1/7], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (3, 5, [1/3, -1/4, 1/5, -1/6, 1/7], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0028/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 29. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=4, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(4, 5, [1/4, -1/5, 1/6, -1/7, 1/8], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (4, 5, [1/4, -1/5, 1/6, -1/7, 1/8], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0029/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 30. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=5, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(5, 5, [1/5, -1/6, 1/7, -1/8, 1/9], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (5, 5, [1/5, -1/6, 1/7, -1/8, 1/9], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0030/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 31. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=6, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(6, 5, [1/6, -1/7, 1/8, -1/9, 1/10], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (6, 5, [1/6, -1/7, 1/8, -1/9, 1/10], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0031/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 32. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=7, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(7, 5, [1/7, -1/8, 1/9, -1/10, 1/11], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (7, 5, [1/7, -1/8, 1/9, -1/10, 1/11], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0032/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 33. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=8, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(8, 5, [1/8, -1/9, 1/10, -1/11, 1/12], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (8, 5, [1/8, -1/9, 1/10, -1/11, 1/12], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0033/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 34. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=9, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(9, 5, [1/9, -1/10, 1/11, -1/12, 1/13], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (9, 5, [1/9, -1/10, 1/11, -1/12, 1/13], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0034/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 35. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=10, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(10, 5, [1/10, -1/11, 1/12, -1/13, 1/14], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (10, 5, [1/10, -1/11, 1/12, -1/13, 1/14], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0035/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 36. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=11, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(11, 5, [1/11, -1/12, 1/13, -1/14, 1/15], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (11, 5, [1/11, -1/12, 1/13, -1/14, 1/15], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0036/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 37. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=12, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(12, 5, [1/12, -1/13, 1/14, -1/15, 1/16], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (12, 5, [1/12, -1/13, 1/14, -1/15, 1/16], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0037/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 38. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=13, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(13, 5, [1/13, -1/14, 1/15, -1/16, 1/17], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (13, 5, [1/13, -1/14, 1/15, -1/16, 1/17], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0038/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 39. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=14, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(14, 5, [1/14, -1/15, 1/16, -1/17, 1/18], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (14, 5, [1/14, -1/15, 1/16, -1/17, 1/18], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0039/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 40. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=15, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(15, 5, [1/15, -1/16, 1/17, -1/18, 1/19], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (15, 5, [1/15, -1/16, 1/17, -1/18, 1/19], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0040/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 41. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=16, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(16, 5, [1/16, -1/17, 1/18, -1/19, 1/20], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (16, 5, [1/16, -1/17, 1/18, -1/19, 1/20], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0041/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 42. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=17, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(17, 5, [1/17, -1/18, 1/19, -1/20, 1/21], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (17, 5, [1/17, -1/18, 1/19, -1/20, 1/21], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0042/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 43. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=18, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(18, 5, [1/18, -1/19, 1/20, -1/21, 1/22], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (18, 5, [1/18, -1/19, 1/20, -1/21, 1/22], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0043/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 44. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=19, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(19, 5, [1/19, -1/20, 1/21, -1/22, 1/23], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (19, 5, [1/19, -1/20, 1/21, -1/22, 1/23], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0044/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 45. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=20, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(20, 5, [1/20, -1/21, 1/22, -1/23, 1/24], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (20, 5, [1/20, -1/21, 1/22, -1/23, 1/24], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0045/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 46. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=21, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(21, 5, [1/21, -1/22, 1/23, -1/24, 1/25], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (21, 5, [1/21, -1/22, 1/23, -1/24, 1/25], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0046/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 47. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=22, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(22, 5, [1/22, -1/23, 1/24, -1/25, 1/26], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (22, 5, [1/22, -1/23, 1/24, -1/25, 1/26], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0047/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 48. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=23, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(23, 5, [1/23, -1/24, 1/25, -1/26, 1/27], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (23, 5, [1/23, -1/24, 1/25, -1/26, 1/27], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0048/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 49. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=24, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(24, 5, [1/24, -1/25, 1/26, -1/27, 1/28], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (24, 5, [1/24, -1/25, 1/26, -1/27, 1/28], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0049/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 50. Conditional Convergence Needs Both Convergence and Absolute Divergence

Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=25, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).

View answer and explanation(25, 5, [1/25, -1/26, 1/27, -1/28, 1/29], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)

The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (25, 5, [1/25, -1/26, 1/27, -1/28, 1/29], Yes, No, alternating_remainder_and_harmonic_absolute_divergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0050/)

Absolute convergence of series · qf-mod35-absseries-conditional-alternating · candidate





## 51. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/2, -1/3]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/2, -1/3], [1/2, 1/3], 5/6, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/2, -1/3], [1/2, 1/3], 5/6, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0051/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 52. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/3, -1/4]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/3, -1/4], [1/3, 1/4], 7/12, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/3, -1/4], [1/3, 1/4], 7/12, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0052/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 53. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/4, -1/5]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/4, -1/5], [1/4, 1/5], 9/20, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/4, -1/5], [1/4, 1/5], 9/20, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0053/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 54. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/5, -1/6]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/5, -1/6], [1/5, 1/6], 11/30, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/5, -1/6], [1/5, 1/6], 11/30, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0054/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 55. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/6, -1/7]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/6, -1/7], [1/6, 1/7], 13/42, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/6, -1/7], [1/6, 1/7], 13/42, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0055/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 56. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/7, -1/8]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/7, -1/8], [1/7, 1/8], 15/56, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/7, -1/8], [1/7, 1/8], 15/56, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0056/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 57. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/8, -1/9]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/8, -1/9], [1/8, 1/9], 17/72, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/8, -1/9], [1/8, 1/9], 17/72, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0057/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 58. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/9, -1/10]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/9, -1/10], [1/9, 1/10], 19/90, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/9, -1/10], [1/9, 1/10], 19/90, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0058/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 59. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/10, -1/11]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/10, -1/11], [1/10, 1/11], 21/110, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/10, -1/11], [1/10, 1/11], 21/110, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0059/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 60. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/11, -1/12]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/11, -1/12], [1/11, 1/12], 23/132, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/11, -1/12], [1/11, 1/12], 23/132, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0060/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 61. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/12, -1/13]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/12, -1/13], [1/12, 1/13], 25/156, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/12, -1/13], [1/12, 1/13], 25/156, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0061/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 62. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/13, -1/14]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/13, -1/14], [1/13, 1/14], 27/182, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/13, -1/14], [1/13, 1/14], 27/182, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0062/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 63. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/14, -1/15]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/14, -1/15], [1/14, 1/15], 29/210, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/14, -1/15], [1/14, 1/15], 29/210, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0063/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 64. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/15, -1/16]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/15, -1/16], [1/15, 1/16], 31/240, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/15, -1/16], [1/15, 1/16], 31/240, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0064/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 65. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/16, -1/17]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/16, -1/17], [1/16, 1/17], 33/272, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/16, -1/17], [1/16, 1/17], 33/272, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0065/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 66. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/17, -1/18]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/17, -1/18], [1/17, 1/18], 35/306, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/17, -1/18], [1/17, 1/18], 35/306, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0066/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 67. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/18, -1/19]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/18, -1/19], [1/18, 1/19], 37/342, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/18, -1/19], [1/18, 1/19], 37/342, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0067/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 68. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/19, -1/20]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/19, -1/20], [1/19, 1/20], 39/380, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/19, -1/20], [1/19, 1/20], 39/380, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0068/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 69. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/20, -1/21]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/20, -1/21], [1/20, 1/21], 41/420, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/20, -1/21], [1/20, 1/21], 41/420, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0069/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 70. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/21, -1/22]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/21, -1/22], [1/21, 1/22], 43/462, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/21, -1/22], [1/21, 1/22], 43/462, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0070/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 71. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/22, -1/23]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/22, -1/23], [1/22, 1/23], 45/506, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/22, -1/23], [1/22, 1/23], 45/506, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0071/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 72. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/23, -1/24]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/23, -1/24], [1/23, 1/24], 47/552, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/23, -1/24], [1/23, 1/24], 47/552, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0072/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 73. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/24, -1/25]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/24, -1/25], [1/24, 1/25], 49/600, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/24, -1/25], [1/24, 1/25], 49/600, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0073/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 74. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/25, -1/26]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/25, -1/26], [1/25, 1/26], 51/650, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/25, -1/26], [1/25, 1/26], 51/650, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0074/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 75. A Finite Sum of Absolute Values Is Not an Infinite Certificate

Supplied ordered exact rational terms=[1/26, -1/27]. Give (terms, absolute_terms, absolute_partial_sum, boundary).

View answer and explanation([1/26, -1/27], [1/26, 1/27], 53/702, finite_absolute_partial_sum_only)

Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/26, -1/27], [1/26, 1/27], 53/702, finite_absolute_partial_sum_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0075/)

Absolute convergence of series · qf-mod35-absseries-absolute-partial · candidate





## 76. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=1 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(1, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (1, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0076/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 77. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=2 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(2, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (2, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0077/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 78. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=3 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(3, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (3, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0078/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 79. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=4 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(4, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (4, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0079/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 80. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=5 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(5, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (5, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0080/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 81. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=6 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(6, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (6, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0081/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 82. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=7 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(7, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (7, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0082/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 83. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=8 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(8, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (8, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0083/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 84. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=9 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(9, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (9, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0084/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 85. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=10 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(10, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (10, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0085/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 86. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=11 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(11, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (11, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0086/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 87. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=12 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(12, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (12, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0087/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 88. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=13 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(13, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (13, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0088/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 89. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=14 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(14, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (14, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0089/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 90. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=15 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(15, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (15, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0090/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 91. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=16 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(16, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (16, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0091/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 92. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=17 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(17, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (17, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0092/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 93. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=18 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(18, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (18, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0093/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 94. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=19 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(19, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (19, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0094/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 95. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=20 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(20, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (20, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0095/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 96. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=21 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(21, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (21, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0096/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 97. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=22 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(22, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (22, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0097/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 98. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=23 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(23, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (23, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0098/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 99. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=24 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(24, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (24, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0099/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate





## 100. Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

Use the shifted-harmonic boundary with positive shift=25 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).

View answer and explanation(25, 5, Yes, No, term_zero_not_absolute_convergence)

The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (25, 5, Yes, No, term_zero_not_absolute_convergence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0100/)

Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate



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