Series convergence tests
Problem 7.80 · easy
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 explanation has not been reviewed yet.
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 |
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.