Series convergence tests
Problem 7.26 · easy
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 \( \text{diverges} \)
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.