Telescoping series
Problem 7.447 · easy
Find the sum of \( \displaystyle \sum_{n=1}^{\infty} \frac{4}{n^{2} + 4 n + 3} \), or show that it diverges.
- \[ \frac{4}{n^{2} + 4 n + 3} = - \frac{2}{n + 3} + \frac{2}{n + 1} \]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.
- So S_N = 5/3 - 2/(N + 3) - 2/(N + 2).
- \[ \lim_{k \to \infty}\left(\frac{5}{3} - \frac{2}{k + 3} - \frac{2}{k + 2}\right) = \frac{5}{3} \]Let N → ∞.✓ Proved
Answer \( \frac{5}{3} \)
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: fail (misleading) — The sentence claims that only the first positive and last negative terms survive, but the second positive term (2/3) also remains in the telescoping sum. This could mislead a student into thinking the cancellation is complete after the first term.qwen3.6:27b-mlx: pass — The solution correctly identifies the telescoping nature of the series and accurately calculates the partial sum and its limit.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-08 — The sentence claims that only the first positive and last negative terms survive, but the second positive term (2/3) also remains in the telescoping sum. This could mislead a student into thinking the cancellation is complete after the first term.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the telescoping nature of the series and accurately calculates the partial sum and its limit.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08
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.