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MathCalculus

Integral Test: Does ∑ 1/[n(ln n)²] Converge?

Check positivity, continuity, and monotonicity on [2,∞), then use u=ln x to prove convergence without equating the sum and integral.

Question

Apply the integral test to decide whether this series converges:

∑n=2∞1n(ln⁡n)2.\sum_{n=2}^{\infty}\frac{1}{n(\ln n)^2}.

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Evidence boundary

This is OpenStax §5.3 exercise 164, beginning at n=2. The integral test proves convergence but does not evaluate the infinite sum. No experimental or student-supplied data are required.