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Nonnegative Tail Comparisons and Correct Convergence Directions
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261015-advance-20261117-r01-series-comparison-test-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=1,B=2, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(1, 2, 1/7, 7/6, 7/3, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (1, 2, 1/7, 7/6, 7/3, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
2. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=2,B=3, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(2, 3, 1/4, 8/3, 4, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (2, 3, 1/4, 8/3, 4, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
3. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=3,B=4, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(3, 4, 1/3, 9/2, 6, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (3, 4, 1/3, 9/2, 6, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
4. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=4,B=5, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(4, 5, 2/5, 20/3, 25/3, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (4, 5, 2/5, 20/3, 25/3, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
5. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=5,B=6, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(5, 6, 5/11, 55/6, 11, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (5, 6, 5/11, 55/6, 11, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
6. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=6,B=7, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(6, 7, 1/7, 7, 49/6, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (6, 7, 1/7, 7, 49/6, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
7. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=7,B=8, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(7, 8, 1/4, 28/3, 32/3, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (7, 8, 1/4, 28/3, 32/3, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
8. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=8,B=9, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(8, 9, 1/3, 12, 27/2, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (8, 9, 1/3, 12, 27/2, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
9. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=9,B=10, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(9, 10, 2/5, 15, 50/3, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (9, 10, 2/5, 15, 50/3, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
10. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=10,B=11, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(10, 11, 5/11, 55/3, 121/6, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (10, 11, 5/11, 55/3, 121/6, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
11. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=11,B=12, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(11, 12, 1/7, 77/6, 14, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (11, 12, 1/7, 77/6, 14, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
12. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=12,B=13, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(12, 13, 1/4, 16, 52/3, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (12, 13, 1/4, 16, 52/3, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
13. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=13,B=14, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(13, 14, 1/3, 39/2, 21, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (13, 14, 1/3, 39/2, 21, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
14. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=14,B=15, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(14, 15, 2/5, 70/3, 25, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (14, 15, 2/5, 70/3, 25, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
15. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=15,B=16, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(15, 16, 5/11, 55/2, 88/3, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (15, 16, 5/11, 55/2, 88/3, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
16. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=16,B=17, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(16, 17, 1/7, 56/3, 119/6, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (16, 17, 1/7, 56/3, 119/6, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
17. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=17,B=18, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(17, 18, 1/4, 68/3, 24, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (17, 18, 1/4, 68/3, 24, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
18. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=18,B=19, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(18, 19, 1/3, 27, 57/2, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (18, 19, 1/3, 27, 57/2, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
19. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=19,B=20, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(19, 20, 2/5, 95/3, 100/3, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (19, 20, 2/5, 95/3, 100/3, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
20. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=20,B=21, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(20, 21, 5/11, 110/3, 77/2, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (20, 21, 5/11, 110/3, 77/2, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
21. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=21,B=22, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(21, 22, 1/7, 49/2, 77/3, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (21, 22, 1/7, 49/2, 77/3, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
22. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=22,B=23, rational ratio r=2/8. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(22, 23, 1/4, 88/3, 92/3, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (22, 23, 1/4, 88/3, 92/3, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
23. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=23,B=24, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(23, 24, 1/3, 69/2, 36, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (23, 24, 1/3, 69/2, 36, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
24. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=24,B=25, rational ratio r=4/10. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(24, 25, 2/5, 40, 125/3, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (24, 25, 2/5, 40, 125/3, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
25. A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series
Use exact scales A=25,B=26, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
View answer and explanation
(25, 26, 5/11, 275/6, 143/3, Yes, upper_comparison_proves_convergence)
Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (25, 26, 5/11, 275/6, 143/3, Yes, upper_comparison_proves_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-upper-geometric · candidate
26. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=2,B=1. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(2, 1, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (2, 1, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
27. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=4,B=2. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(4, 2, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (4, 2, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
28. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=6,B=3. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(6, 3, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (6, 3, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
29. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=8,B=4. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(8, 4, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (8, 4, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
30. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=10,B=5. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(10, 5, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (10, 5, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
31. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=12,B=6. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(12, 6, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (12, 6, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
32. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=14,B=7. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(14, 7, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (14, 7, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
33. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=16,B=8. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(16, 8, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (16, 8, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
34. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=18,B=9. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(18, 9, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (18, 9, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
35. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=20,B=10. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(20, 10, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (20, 10, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
36. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=22,B=11. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(22, 11, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (22, 11, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
37. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=24,B=12. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(24, 12, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (24, 12, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
38. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=26,B=13. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(26, 13, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (26, 13, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
39. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=28,B=14. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(28, 14, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (28, 14, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
40. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=30,B=15. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(30, 15, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (30, 15, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
41. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=32,B=16. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(32, 16, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (32, 16, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
42. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=34,B=17. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(34, 17, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (34, 17, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
43. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=36,B=18. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(36, 18, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (36, 18, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
44. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=38,B=19. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(38, 19, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (38, 19, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
45. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=40,B=20. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(40, 20, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (40, 20, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
46. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=42,B=21. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(42, 21, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (42, 21, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
47. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=44,B=22. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(44, 22, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (44, 22, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
48. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=46,B=23. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(46, 23, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (46, 23, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
49. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=48,B=24. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(48, 24, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (48, 24, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
50. A Positive Constant Lower Bound Gives Unbounded Partial Sums
Use exact scales A=50,B=25. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).
View answer and explanation
(50, 25, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)
The reference partial sums with N terms equal N*B and are unbounded. The larger series therefore diverges. Preserve the lower-bound direction and do not infer divergence from a divergent upper bound. The exact result, retaining the requested order, is (50, 25, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-lower-divergence · candidate
51. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=1,B=2, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(0, 0, 0, 8/7, 16/7, 8/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 8/7, 16/7, 8/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
52. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=2,B=3, rational ratio r=2/9. Prefix a=[2], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(1, 2, 1, 18/7, 27/7, 32/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 2, 1, 18/7, 27/7, 32/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
53. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=3,B=4, rational ratio r=3/10. Prefix a=[3, 4], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(2, 7, 3, 30/7, 40/7, 79/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 7, 3, 30/7, 40/7, 79/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
54. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=4,B=5, rational ratio r=4/11. Prefix a=[4, 5, 6], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(3, 15, 6, 44/7, 55/7, 149/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 15, 6, 44/7, 55/7, 149/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
55. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=5,B=6, rational ratio r=5/12. Prefix a=[5, 6, 7, 8], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(4, 26, 10, 60/7, 72/7, 242/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 26, 10, 60/7, 72/7, 242/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
56. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=6,B=7, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(0, 0, 0, 48/7, 8, 48/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 48/7, 8, 48/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
57. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=7,B=8, rational ratio r=2/9. Prefix a=[7], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(1, 7, 1, 9, 72/7, 16, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 7, 1, 9, 72/7, 16, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
58. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=8,B=9, rational ratio r=3/10. Prefix a=[8, 9], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(2, 17, 3, 80/7, 90/7, 199/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 17, 3, 80/7, 90/7, 199/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
59. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=9,B=10, rational ratio r=4/11. Prefix a=[9, 10, 11], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(3, 30, 6, 99/7, 110/7, 309/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 30, 6, 99/7, 110/7, 309/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
60. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=10,B=11, rational ratio r=5/12. Prefix a=[10, 11, 12, 13], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(4, 46, 10, 120/7, 132/7, 442/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 46, 10, 120/7, 132/7, 442/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
61. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=11,B=12, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(0, 0, 0, 88/7, 96/7, 88/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 88/7, 96/7, 88/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
62. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=12,B=13, rational ratio r=2/9. Prefix a=[12], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(1, 12, 1, 108/7, 117/7, 192/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 12, 1, 108/7, 117/7, 192/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
63. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=13,B=14, rational ratio r=3/10. Prefix a=[13, 14], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(2, 27, 3, 130/7, 20, 319/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 27, 3, 130/7, 20, 319/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
64. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=14,B=15, rational ratio r=4/11. Prefix a=[14, 15, 16], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(3, 45, 6, 22, 165/7, 67, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 45, 6, 22, 165/7, 67, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
65. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=15,B=16, rational ratio r=5/12. Prefix a=[15, 16, 17, 18], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(4, 66, 10, 180/7, 192/7, 642/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 66, 10, 180/7, 192/7, 642/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
66. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=16,B=17, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(0, 0, 0, 128/7, 136/7, 128/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 128/7, 136/7, 128/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
67. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=17,B=18, rational ratio r=2/9. Prefix a=[17], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(1, 17, 1, 153/7, 162/7, 272/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 17, 1, 153/7, 162/7, 272/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
68. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=18,B=19, rational ratio r=3/10. Prefix a=[18, 19], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(2, 37, 3, 180/7, 190/7, 439/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 37, 3, 180/7, 190/7, 439/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
69. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=19,B=20, rational ratio r=4/11. Prefix a=[19, 20, 21], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(3, 60, 6, 209/7, 220/7, 629/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 60, 6, 209/7, 220/7, 629/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
70. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=20,B=21, rational ratio r=5/12. Prefix a=[20, 21, 22, 23], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(4, 86, 10, 240/7, 36, 842/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 86, 10, 240/7, 36, 842/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
71. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=21,B=22, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(0, 0, 0, 24, 176/7, 24, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (0, 0, 0, 24, 176/7, 24, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
72. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=22,B=23, rational ratio r=2/9. Prefix a=[22], prefix b=[1], each with cutoff=1 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(1, 22, 1, 198/7, 207/7, 352/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (1, 22, 1, 198/7, 207/7, 352/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
73. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=23,B=24, rational ratio r=3/10. Prefix a=[23, 24], prefix b=[1, 2], each with cutoff=2 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(2, 47, 3, 230/7, 240/7, 559/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (2, 47, 3, 230/7, 240/7, 559/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
74. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=24,B=25, rational ratio r=4/11. Prefix a=[24, 25, 26], prefix b=[1, 2, 3], each with cutoff=3 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(3, 75, 6, 264/7, 275/7, 789/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (3, 75, 6, 264/7, 275/7, 789/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
75. Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails
Use exact scales A=25,B=26, rational ratio r=5/12. Prefix a=[25, 26, 27, 28], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes.
View answer and explanation
(4, 106, 10, 300/7, 312/7, 1042/7, Yes, finite_prefix_does_not_change_tail_convergence)
Add each finite prefix separately and compute both geometric tails. The tail comparison does not require the prefixes or total sums to have the same ordering. The convergence flag concerns the full valid tail, not finitely sampled terms. The exact result, retaining the requested order, is (4, 106, 10, 300/7, 312/7, 1042/7, Yes, finite_prefix_does_not_change_tail_convergence).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-eventual-tail · candidate
76. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=finite_sample, positive scale=1. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(finite_sample, 1, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 1, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
77. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=divergent_upper, positive scale=2. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(divergent_upper, a_n=2/n^2, b_n=2/n, a_n<=b_n, b_diverges_but_a_converges, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=2/n^2, b_n=2/n, a_n<=b_n, b_diverges_but_a_converges, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
78. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=signed_terms, positive scale=3. Give (mode, series_description, failed_hypothesis, comparison_test_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(signed_terms, a_n=(-1)^n*3/n, nonnegative_hypothesis_fails, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*3/n, nonnegative_hypothesis_fails, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
79. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=unknown_reference, positive scale=4. Give (mode, scale, reference_status_scope, comparison_test_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(unknown_reference, 4, comparison_holds_reference_status_not_proved, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 4, comparison_holds_reference_status_not_proved, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
80. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=finite_sample, positive scale=5. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(finite_sample, 5, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 5, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
81. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=divergent_upper, positive scale=6. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(divergent_upper, a_n=6/n^2, b_n=6/n, a_n<=b_n, b_diverges_but_a_converges, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=6/n^2, b_n=6/n, a_n<=b_n, b_diverges_but_a_converges, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
82. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=signed_terms, positive scale=7. Give (mode, series_description, failed_hypothesis, comparison_test_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(signed_terms, a_n=(-1)^n*7/n, nonnegative_hypothesis_fails, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*7/n, nonnegative_hypothesis_fails, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
83. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=unknown_reference, positive scale=8. Give (mode, scale, reference_status_scope, comparison_test_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(unknown_reference, 8, comparison_holds_reference_status_not_proved, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 8, comparison_holds_reference_status_not_proved, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
84. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=finite_sample, positive scale=9. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(finite_sample, 9, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 9, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
85. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=divergent_upper, positive scale=10. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(divergent_upper, a_n=10/n^2, b_n=10/n, a_n<=b_n, b_diverges_but_a_converges, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=10/n^2, b_n=10/n, a_n<=b_n, b_diverges_but_a_converges, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
86. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=signed_terms, positive scale=11. Give (mode, series_description, failed_hypothesis, comparison_test_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(signed_terms, a_n=(-1)^n*11/n, nonnegative_hypothesis_fails, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*11/n, nonnegative_hypothesis_fails, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
87. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=unknown_reference, positive scale=12. Give (mode, scale, reference_status_scope, comparison_test_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(unknown_reference, 12, comparison_holds_reference_status_not_proved, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 12, comparison_holds_reference_status_not_proved, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
88. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=finite_sample, positive scale=13. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(finite_sample, 13, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 13, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
89. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=divergent_upper, positive scale=14. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(divergent_upper, a_n=14/n^2, b_n=14/n, a_n<=b_n, b_diverges_but_a_converges, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=14/n^2, b_n=14/n, a_n<=b_n, b_diverges_but_a_converges, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
90. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=signed_terms, positive scale=15. Give (mode, series_description, failed_hypothesis, comparison_test_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(signed_terms, a_n=(-1)^n*15/n, nonnegative_hypothesis_fails, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*15/n, nonnegative_hypothesis_fails, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
91. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=unknown_reference, positive scale=16. Give (mode, scale, reference_status_scope, comparison_test_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(unknown_reference, 16, comparison_holds_reference_status_not_proved, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 16, comparison_holds_reference_status_not_proved, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
92. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=finite_sample, positive scale=17. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(finite_sample, 17, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 17, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
93. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=divergent_upper, positive scale=18. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(divergent_upper, a_n=18/n^2, b_n=18/n, a_n<=b_n, b_diverges_but_a_converges, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=18/n^2, b_n=18/n, a_n<=b_n, b_diverges_but_a_converges, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
94. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=signed_terms, positive scale=19. Give (mode, series_description, failed_hypothesis, comparison_test_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(signed_terms, a_n=(-1)^n*19/n, nonnegative_hypothesis_fails, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*19/n, nonnegative_hypothesis_fails, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
95. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=unknown_reference, positive scale=20. Give (mode, scale, reference_status_scope, comparison_test_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(unknown_reference, 20, comparison_holds_reference_status_not_proved, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 20, comparison_holds_reference_status_not_proved, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
96. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=finite_sample, positive scale=21. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(finite_sample, 21, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 21, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
97. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=divergent_upper, positive scale=22. Give (mode, series_a_description, series_b_description, term_comparison, actual_series_status, divergence_inference_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(divergent_upper, a_n=22/n^2, b_n=22/n, a_n<=b_n, b_diverges_but_a_converges, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (divergent_upper, a_n=22/n^2, b_n=22/n, a_n<=b_n, b_diverges_but_a_converges, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
98. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=signed_terms, positive scale=23. Give (mode, series_description, failed_hypothesis, comparison_test_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(signed_terms, a_n=(-1)^n*23/n, nonnegative_hypothesis_fails, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (signed_terms, a_n=(-1)^n*23/n, nonnegative_hypothesis_fails, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
99. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=unknown_reference, positive scale=24. Give (mode, scale, reference_status_scope, comparison_test_claim).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(unknown_reference, 24, comparison_holds_reference_status_not_proved, No)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (unknown_reference, 24, comparison_holds_reference_status_not_proved, No).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
100. Comparison Direction, Nonnegativity and Reference Status Cannot Be Omitted
Boundary mode=finite_sample, positive scale=25. Give (mode, scale, sample_scope, eventual_comparison_claim, boundary).
The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof.
View answer and explanation
(finite_sample, 25, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall)
finite_sample records insufficient eventual-forall evidence. A scaled1/n^2 series can converge below divergent scaled1/n; signed terms violate the nonnegative premise; an unproved reference status supplies no convergence certificate. Keep the distinct branch field roles, not one padded tuple. The exact result, retaining the requested order, is (finite_sample, 25, first_N_terms_compare_next_term_fails, No, finite_samples_do_not_establish_eventual_forall).
Reviewed step-by-step derivation
Comparison tests for series · qf-mod48-series-comparison-boundary · candidate
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