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Series convergence tests

Problem 7.90 · easy Video

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{4 n + 1}{n^{2} + n + 1} \) converge or diverge?

For large n the terms behave like 4/n^1: compare with the p-series Σ 1/n^1.
Reviewed
\[ \lim_{n \to \infty}\left(\frac{n \left(4 n + 1\right)}{n^{2} + n + 1}\right) = 4 \]
The limit is 4, positive and finite.✓ Proved
Σ 1/n^1 diverges (p = 1), so the given series does too.
Reviewed

Answer \( \text{diverges} \)

Bars show the terms (4n + 1)/(n² + n + 1) for n = 1 to 24; dots show the running totals. The totals never settle: the series diverges.

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	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 applies the Limit Comparison Test with the harmonic series, which is a valid method for determining divergence.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the Limit Comparison Test with the harmonic series, which is a valid method for determining divergence.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the Limit Comparison Test with the divergent harmonic series. The limit calculation is correct, and the conclusion follows logically.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the Limit Comparison Test with the harmonic series, which is the standard and rigorous method for this problem.
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 2026-09-28 with SymPy 1.14.0.