Telescoping series
Problem 7.446 · easy
Find the sum of \( \displaystyle \sum_{n=1}^{\infty} \frac{3}{n^{2} + 4 n + 3} \), or show that it diverges.
- \[ \frac{3}{n^{2} + 4 n + 3} = - \frac{3}{2 n + 6} + \frac{3}{2 n + 2} \]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.Reviewed
- So S_N = 5/4 - 3/(2*(N + 3)) - 3/(2*(N + 2)).Reviewed
- \[ \lim_{k \to \infty}\left(\frac{5}{4} - \frac{3}{2 k + 6} - \frac{3}{2 k + 4}\right) = \frac{5}{4} \]Let N → ∞.✓ Proved
Answer \( \frac{5}{4} \)
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 determines the surviving terms in the partial sum, and correctly evaluates the limit.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the telescoping nature of the series, accurately determines the surviving terms in the partial sum, and correctly evaluates the limit.gpt-oss:20b: fail (error) 2026-10-08 — The cancellation argument is incorrect: the first two positive terms (n=1 and n=2) survive, not just the first one. Consequently the expression for S_N is missing the +1/2 term, leading to an incorrect limit. The correct partial sum is S_N=5/4-3/(2(N+2))-3/(2(N+3)).qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the telescoping nature of the series and accurately accounts for the two surviving terms in the partial sum. The limit calculation is correct.
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.