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Series convergence tests

Problem 7.26 · easy Video

Does \( \displaystyle \sum_{n=2}^{\infty} \frac{1}{n \ln{\left(n \right)}} \) converge or diverge?

f(x) = 1/(x (ln x)^p) is positive, continuous and decreasing for x ≥ 2: use the integral test.
\[ \frac{d}{d x} \ln{\left(\ln{\left(x \right)} \right)} = \frac{1}{x \ln{\left(x \right)}} \]
An antiderivative of f.✓ Proved
The improper integral from 2 to ∞ diverges (ln(ln x) → ∞), so the series diverges.

Answer \( \text{diverges} \)

Bars show the terms 1/(n·log(n)) for n = 2 to 25; dots show the running totals. The totals never settle: the series diverges.

Lines: 1 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	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: fail (error) — The solution defines f(x) with an arbitrary parameter p, but the problem specifies p=1. While the conclusion is correct, the setup is imprecise and potentially confusing because it introduces a parameter not present in the specific problem statement.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution defines f(x) with an arbitrary parameter p, but the problem specifies p=1. While the conclusion is correct, the setup is imprecise and potentially confusing because it introduces a parameter not present in the specific problem statement.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution defines f(x) with an arbitrary parameter p, but the problem specifies p=1. While the conclusion is correct, the setup is imprecise and potentially confusing regarding the specific function being tested.
gpt-oss:20b: fail (misleading) 2026-09-27 — The solution introduces a parameter p in the function f(x)=1/(x(ln x)^p) but never specifies that p=1, which is required for the given series. This omission could mislead a reader into thinking the result holds for any p, whereas the conclusion only applies to p=1.
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution defines f(x) with an arbitrary parameter p, but the problem specifies p=1. While the conclusion is correct, the setup is imprecise and potentially confusing regarding the specific function being tested.
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.