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Series convergence tests

Problem 7.162 · easy

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{5 n + 1}{5 n + 3} \) converge or diverge?

\[ \lim_{n \to \infty}\left(\frac{5 n + 1}{5 n + 3}\right) = 1 \]
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 — The solution correctly identifies that the limit of the terms is 1, not 0, and correctly applies the Divergence Test to conclude the series diverges.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies that the limit of the terms is 1, not 0, and correctly applies the Divergence Test to conclude the series diverges.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the Divergence Test (nth term test). The limit of the terms is 1, which is not 0, so the series diverges.
gpt-oss:20b: pass 2026-09-27

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 2026-09-27 with SymPy 1.14.0.