Telescoping series
Problem 7.497 · easy
Find the sum of \( \displaystyle \sum_{n=1}^{\infty} \frac{6}{n^{2} + 3 n} \), or show that it diverges.
- \[ \frac{6}{n^{2} + 3 n} = - \frac{2}{n + 3} + \frac{2}{n} \]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 = 11/3 - 2/(N + 3) - 2/(N + 2) - 2/(N + 1).Reviewed
- \[ \lim_{k \to \infty}\left(\frac{11}{3} - \frac{2}{k + 3} - \frac{2}{k + 2} - \frac{2}{k + 1}\right) = \frac{11}{3} \]Let N → ∞.✓ Proved
Answer \( \frac{11}{3} \)
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. The partial sum formula and the limit are correct.
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 with a shift of 3. The partial sum formula and the limit are correct.gpt-oss:20b: fail (error) 2026-10-10 — The cancellation argument is incorrect: the last three positive terms (n=N‑2,N‑1,N) do not cancel and must be retained. The resulting partial sum should include +2/(N‑2)+2/(N‑1)+2/N, not just the first positive and last negative terms.qwen3.6:27b-mlx: fail (error) 2026-10-10 — The partial fraction decomposition in line 1 is incorrect; the correct decomposition is 2/n - 2/(n+3). Consequently, the partial sum formula in line 3 is wrong (the constant term should be 11/3, but the signs of the remaining terms are flipped, and the logic in line 2 is imprecise regarding which terms survive).
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.