Telescoping series
Problem 7.449 · easy
Find the sum of \( \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{2} + 8 n + 15} \), or show that it diverges.
- \[ \frac{1}{n^{2} + 8 n + 15} = - \frac{1}{2 n + 10} + \frac{1}{2 n + 6} \]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.
- So S_N = 9/40 - 1/(2*(N + 5)) - 1/(2*(N + 4)).
- \[ \lim_{k \to \infty}\left(\frac{9}{40} - \frac{1}{2 k + 10} - \frac{1}{2 k + 8}\right) = \frac{9}{40} \]Let N → ∞.✓ Proved
Answer \( \frac{9}{40} \)
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 is incorrect: the positive term 1/(2(n+3)) does not cancel with a negative term two indices later, so the expression for S_N is wrong.qwen3.6:27b-mlx: pass — The solution correctly applies partial fraction decomposition and identifies the telescoping nature of the series. The calculation of the partial sum limit is accurate.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-08 — The cancellation argument is incorrect: the positive term 1/(2(n+3)) does not cancel with a negative term two indices later, so the expression for S_N is wrong.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies partial fraction decomposition and identifies the telescoping nature of the series. The calculation of the partial sum limit is accurate.gpt-oss:20b: fail (misleading) 2026-10-08 — The sentence claiming that each negative term cancels a positive term two indices later is incorrect; the cancellation occurs between the positive part of term n and the negative part of term n+2, but the description is misleading and could confuse a student.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the telescoping nature of the series and accurately calculates the limit of the partial sums.
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.