Telescoping series
Problem 7.453 · easy
Find the sum of \( \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{2} + 9 n + 18} \), or show that it diverges.
- \[ \frac{1}{n^{2} + 9 n + 18} = - \frac{1}{3 n + 18} + \frac{1}{3 n + 9} \]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 = 37/180 - 1/(3*(N + 6)) - 1/(3*(N + 5)) - 1/(3*(N + 4)).
- \[ \lim_{k \to \infty}\left(\frac{37}{180} - \frac{1}{3 k + 18} - \frac{1}{3 k + 15} - \frac{1}{3 k + 12}\right) = \frac{37}{180} \]Let N → ∞.✓ Proved
Answer \( \frac{37}{180} \)
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 (misleading) — The description of the cancellation pattern in sentence 2 is incorrect; the positive term at index n does not cancel a negative term three indices later. The telescoping argument is valid, but the wording misleads a student about how the terms cancel.qwen3.6:27b-mlx: pass — The solution correctly identifies the telescoping nature of the series with a shift of 3. The partial sum formula and the limit are correct.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-08 — The description of the cancellation pattern in sentence 2 is incorrect; the positive term at index n does not cancel a negative term three indices later. The telescoping argument is valid, but the wording misleads a student about how the terms cancel.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the telescoping nature of the series with a shift of 3. The partial sum formula and the limit are correct.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the telescoping nature of the series with a shift of 3. The partial sum formula and the limit calculation are correct, leading to the right answer.
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.