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Telescoping series

Problem 7.321 · easy

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

\[ \frac{5}{n^{2} + 4 n + 3} = - \frac{5}{2 n + 6} + \frac{5}{2 n + 2} \]
Partial fractions: the term is a difference.✓ Proved
In the N-th partial sum every negative piece cancels a positive piece 2 terms later; only the first positive and last negative pieces survive.
Reviewed
So S_N = 25/12 - 5/(2*(N + 3)) - 5/(2*(N + 2)).
Reviewed
\[ \lim_{k \to \infty}\left(\frac{25}{12} - \frac{5}{2 k + 6} - \frac{5}{2 k + 4}\right) = \frac{25}{12} \]
Let N → ∞.✓ Proved

Answer \( \frac{25}{12} \)

✓ 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 partial sum by accounting for the two surviving terms, and correctly takes the limit to find the sum.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the telescoping nature of the series, accurately calculates the partial sum by accounting for the two surviving terms, and correctly takes the limit to find the sum.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the telescoping nature of the series, accurately calculates the partial sum formula, and takes the correct limit to find the sum.
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.