Home›Calculus 2›Limits of sequences›Problem 7.382

Limits of sequences

Problem 7.382 · medium

Write the first four terms of \( \displaystyle a_n = \frac{\left(-1\right)^{n} n}{n + 1} \), and determine whether the sequence converges. If it does, find its limit.

\[ \left[\begin{matrix}- \frac{1}{2}\\\frac{2}{3}\\- \frac{3}{4}\\\frac{4}{5}\end{matrix}\right] \]
The first four terms.✓ Proved
The terms approach 1 for even n and −1 for odd n.
\[ \lim_{n \to \infty}\left(\frac{2 n}{2 n + 1}\right) = 1 \]
Even-numbered terms tend to 1 ...✓ Proved
\[ \lim_{n \to \infty}\left(\frac{- 2 n - 1}{2 n + 2}\right) = -1 \]
... odd-numbered terms to −1, so there is no single limit.✓ Proved

Answer \( a_1, \dots, a_4 = - \frac{1}{2}, \frac{2}{3}, - \frac{3}{4}, \frac{4}{5};\ \text{diverges} \)

Lines: 3 proved, 1 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
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	the terms evaluated at n = 10⁶, 10⁶ + 1 and 10⁷

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly concludes that the sequence diverges. The terms actually approach -1 for both even and odd n (since (-1)^n * n/(n+1) -> -1 as n->oo), so the sequence converges to -1. The error stems from ignoring the (-1)^n factor in the limit calculation for even n, treating the term as positive n/(n+1) instead of -n/(n+1).

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution incorrectly concludes that the sequence diverges. The terms actually approach -1 for both even and odd n (since (-1)^n * n/(n+1) -> -1 as n->oo), so the sequence converges to -1. The error stems from ignoring the (-1)^n factor in the limit calculation for even n, treating the term as positive n/(n+1) instead of -n/(n+1).
qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly concludes that the sequence diverges. The terms actually approach -1 for both even and odd n (since (-1)^n * n/(n+1) -> (-1
gpt-oss:20b: pass 2026-10-06

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/sequence_limits, checked 2026-10-06 with SymPy 1.14.0.