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Telescoping series

Problem 7.454 · easy

Find the sum of \( \displaystyle \sum_{n=1}^{\infty} \frac{3}{n^{2} + 5 n + 4} \), or show that it diverges.

\[ \frac{3}{n^{2} + 5 n + 4} = - \frac{1}{n + 4} + \frac{1}{n + 1} \]
Partial fractions: the term is a difference.✓ Proved
In the N-th partial sum every negative piece cancels a positive piece 3 terms later; only the first positive and last negative pieces survive.
Reviewed
So S_N = 13/12 - 1/(N + 4) - 1/(N + 3) - 1/(N + 2).
Reviewed
\[ \lim_{k \to \infty}\left(\frac{13}{12} - \frac{1}{k + 4} - \frac{1}{k + 3} - \frac{1}{k + 2}\right) = \frac{13}{12} \]
Let N → ∞.✓ Proved

Answer \( \frac{13}{12} \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	mpmath's nsum of the series (with convergence acceleration)

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the telescoping nature of the series, accurately calculates the surviving terms of the partial sum, and correctly evaluates the limit.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the telescoping nature of the series, accurately calculates the surviving terms of the partial sum, and correctly evaluates the limit.
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: pass 2026-10-08

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/telescoping_series, checked 2026-10-08 with SymPy 1.14.0.