Telescoping series
Problem 7.398 · easy
Find the sum of \( \displaystyle \sum_{n=1}^{\infty} \frac{2}{n^{2} + 9 n + 18} \), or show that it diverges.
- \[ \frac{2}{n^{2} + 9 n + 18} = - \frac{2}{3 n + 18} + \frac{2}{3 n + 9} \]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 = 37/90 - 2/(3*(N + 6)) - 2/(3*(N + 5)) - 2/(3*(N + 4)).
- \[ \lim_{k \to \infty}\left(\frac{37}{90} - \frac{2}{3 k + 18} - \frac{2}{3 k + 15} - \frac{2}{3 k + 12}\right) = \frac{37}{90} \]Let N → ∞.✓ Proved
Answer \( \frac{37}{90} \)
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: passqwen3.6:27b-mlx: fail (error) — The partial fraction decomposition is incorrect; the correct decomposition is 2/(n^2 + 9n + 18) = 2/(3(n+3)) - 2/(3(n+6)). The solution has the signs reversed, leading to a divergent series (sum of positive terms treated as negative) and an incorrect final answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The partial fraction decomposition is incorrect; the correct decomposition is 2/(n^2 + 9n + 18) = 2/(3(n+3)) - 2/(3(n+6)). The solution has the signs reversed, leading to a divergent series (sum of positive terms treated as negative) and an incorrect final answer.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The partial fraction decomposition is incorrect; the correct decomposition yields coefficients of 2/3 and -2/3, not -2/3 and 2/3 as implied by the signs in the equation. Consequently, the telescoping sum calculation and the final result are wrong.
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.