Telescoping series
Problem 7.543 · easy
Find the sum of \( \displaystyle \sum_{n=1}^{\infty} \frac{3}{n^{2} + 9 n + 18} \), or show that it diverges.
- \[ \frac{3}{n^{2} + 9 n + 18} = - \frac{1}{n + 6} + \frac{1}{n + 3} \]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 = 37/60 - 1/(N + 6) - 1/(N + 5) - 1/(N + 4).
- \[ \lim_{k \to \infty}\left(\frac{37}{60} - \frac{1}{k + 6} - \frac{1}{k + 5} - \frac{1}{k + 4}\right) = \frac{37}{60} \]Let N → ∞.✓ Proved
Answer \( \frac{37}{60} \)
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: fail (error) — The cancellation argument in sentence 2 is incorrect: the terms −1/(n+6) and 1/(n+3) do not cancel three indices apart, so the expression for S_N is not valid.qwen3.6:27b-mlx: pass — The solution correctly identifies the telescoping nature of the series and accurately calculates the surviving terms for the partial sum. The limit calculation is correct.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-13 — The cancellation argument in sentence 2 is incorrect: the terms −1/(n+6) and 1/(n+3) do not cancel three indices apart, so the expression for S_N is not valid.qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the telescoping nature of the series and accurately calculates the surviving terms for the partial sum. The limit calculation is correct.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the telescoping nature of the series, accurately calculates the surviving terms for the partial sum, and correctly evaluates the limit.
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 2026-10-13 with SymPy 1.14.0.