Telescoping series
Problem 7.492 · easy
Find the sum of \( \displaystyle \sum_{n=1}^{\infty} \frac{2}{n^{2} + 5 n + 4} \), or show that it diverges.
- \[ \frac{2}{n^{2} + 5 n + 4} = - \frac{2}{3 n + 12} + \frac{2}{3 n + 3} \]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.Reviewed
- So S_N = 13/18 - 2/(3*(N + 4)) - 2/(3*(N + 3)) - 2/(3*(N + 2)).Reviewed
- \[ \lim_{k \to \infty}\left(\frac{13}{18} - \frac{2}{3 k + 12} - \frac{2}{3 k + 9} - \frac{2}{3 k + 6}\right) = \frac{13}{18} \]Let N → ∞.✓ Proved
Answer \( \frac{13}{18} \)
Lines: 2 proved, 2 reviewed. 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 | 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the telescoping nature of the series, accurately accounts for the 3-term shift in cancellation, and correctly calculates the limit of the partial sums.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the telescoping nature of the series, accurately accounts for the 3-term shift in cancellation, and correctly calculates the limit of the partial sums.gpt-oss:20b: fail (error) 2026-10-10 — The cancellation argument is incorrect: the negative term -2/(3(n+4)) does not cancel a positive term three indices later. The partial‑fraction decomposition is correct, but the subsequent claim that only the first positive and last negative pieces survive is false, so the derived sum is not justified.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the telescoping nature of the series with a lag of 3 terms 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 2026-10-10 with SymPy 1.14.0.