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Series convergence tests

Problem 7.131 · easy

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{n + 1}{n + 3} \) converge or diverge?

\[ \lim_{n \to \infty}\left(\frac{n + 1}{n + 3}\right) = 1 \]
The terms do not go to 0...✓ Proved
...so by the divergence test the series diverges.
Reviewed

Answer \( \text{diverges} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	SymPy's own convergence decision (Sum.is_convergent) agrees

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: pass 2026-09-26

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/series_convergence, checked 2026-09-26 with SymPy 1.14.0.