Series convergence tests
Problem 7.80 · easy Video
Does \( \displaystyle \sum_{n=2}^{\infty} \frac{1}{n \ln{\left(n \right)}^{2}} \) converge or diverge?
- f(x) = 1/(x (ln x)^p) is positive, continuous and decreasing for x ≥ 2: use the integral test.
- \[ \frac{d}{d x} \left(- \frac{1}{\ln{\left(x \right)}}\right) = \frac{1}{x \ln{\left(x \right)}^{2}} \]An antiderivative of f.✓ Proved
- The improper integral from 2 to ∞ converges (−1/ln x → 0), so the series converges.
Answer \( \text{converges} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to specify the value of p in the function definition, making the setup ambiguous. It then incorrectly claims that -1/log(x) is an antiderivative of f(x) without showing that p=2, which is required for the specific series given.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to specify the value of p in the function definition, making the setup ambiguous. It then incorrectly claims that -1/log(x) is an antiderivative of f(x) without showing that p=2, which is required for the specific series given.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution incorrectly identifies the exponent p as 2 in the general form, but the derivative check in line 2 corresponds to p=1 (since d/dx(-1/ln x) = 1/(x ln x)). For p=1, the integral diverges, so the series diverges. The stated answer 'converges' is incorrect for the given series.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the function properties required for the Integral Test, computes the correct antiderivative, and properly evaluates the limit at infinity to conclude convergence.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.