Absolute Geometric Sums, Conditional Harmonic Tails and Finite Boundaries
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Absolute convergence of series · qf-mod35-absseries-term-zero-boundary · candidate
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