Telescoping series
Problem 7.394 · easy
Find the sum of \( \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{2} + 7 n + 10} \), or show that it diverges.
- \[ \frac{1}{n^{2} + 7 n + 10} = - \frac{1}{3 n + 15} + \frac{1}{3 n + 6} \]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 = 47/180 - 1/(3*(N + 5)) - 1/(3*(N + 4)) - 1/(3*(N + 3)).Reviewed
- \[ \lim_{k \to \infty}\left(\frac{47}{180} - \frac{1}{3 k + 15} - \frac{1}{3 k + 12} - \frac{1}{3 k + 9}\right) = \frac{47}{180} \]Let N → ∞.✓ Proved
Answer \( \frac{47}{180} \)
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 with a shift of 3 terms. The constant term 47/180 is derived correctly from the surviving initial terms, and the limit calculation is accurate.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the telescoping nature of the series with a shift of 3 terms. The constant term 47/180 is derived correctly from the surviving initial terms, and the limit calculation is accurate.gpt-oss:20b: fail (error) 2026-10-07 — The cancellation argument is incorrect: the negative term -1/(3(n+5)) does not cancel a positive term three indices later; the partial sums do not telescope as claimed. The conclusion that only the first positive and last negative terms survive is false, so the solution is invalid.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The partial fraction decomposition in line 1 is incorrect; the correct decomposition is 1/3(1/(n+2) - 1/(n+5)). Consequently, the partial sum formula in line 3 is wrong (it should involve 1/(N+2) and 1/(N+3) terms, not the stated ones), although the final limit happens to be correct by coincidence.
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 2026-10-07 with SymPy 1.14.0.