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Telescoping series

Problem 7.326 · easy

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

\[ \frac{4}{n^{2} + 3 n} = - \frac{4}{3 n + 9} + \frac{4}{3 n} \]
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.
So S_N = 22/9 - 4/(3*(N + 3)) - 4/(3*(N + 2)) - 4/(3*(N + 1)).
\[ \lim_{k \to \infty}\left(\frac{22}{9} - \frac{4}{3 k + 9} - \frac{4}{3 k + 6} - \frac{4}{3 k + 3}\right) = \frac{22}{9} \]
Let N → ∞.✓ Proved

Answer \( \frac{22}{9} \)

Lines: 2 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	a sentence; read, not computed
3	Not checked	—	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: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The partial fraction decomposition is incorrect. The correct decomposition of 4/(n(n+3)) is 4/3 * (1/n - 1/(n+3)), but the solution writes -4/(3(n+3)

Every verdict on record (4)

qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The partial fraction decomposition is incorrect. The correct decomposition of 4/(n(n+3)) is 4/3 * (1/n - 1/(n+3)), but the solution writes -4/(3(n+3)
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 with a lag of 3 terms. The partial sum formula and the limit are correct, leading to the right final answer.
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.