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Telescoping series

Problem 7.328 · easy

Find the sum of \( \displaystyle \sum_{n=1}^{\infty} \frac{2}{n^{2} + 2 n} \), or show that it diverges.

\[ \frac{2}{n^{2} + 2 n} = - \frac{1}{n + 2} + \frac{1}{n} \]
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 = 3/2 - 1/(N + 2) - 1/(N + 1).
\[ \lim_{k \to \infty}\left(\frac{3}{2} - \frac{1}{k + 2} - \frac{1}{k + 1}\right) = \frac{3}{2} \]
Let N → ∞.✓ Proved

Answer \( \frac{3}{2} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The partial fraction decomposition is incorrect; the correct decomposition is 1/n - 1/(n+1), not 1/n - 1/(n+2). Consequently, the telescoping pattern and the final sum are wrong (the correct sum is 1).

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The partial fraction decomposition is incorrect; the correct decomposition is 1/n - 1/(n+1), not 1/n - 1/(n+2). Consequently, the telescoping pattern and the final sum are wrong (the correct sum is 1).
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The partial fraction decomposition in step 1 is incorrect; the correct decomposition is 1/n - 1/(n+1), not 1/n - 1/(n+2). Consequently, the telescoping sum and final result are wrong.
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.