Consider the series

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the series ∑n=2∞(−1)nln⁡nn\displaystyle \sum_{n=2}^{\infty} (-1)^{n}\dfrac{\ln n}{n}. Which statement best describes its convergence? (a) It converges absolutely; for large nn, ln⁡nn<1n1.1\dfrac{\ln n}{n}<\dfrac{1}{n^{1.1}}, and the pp-series with p=1.1>1p=1.1>1 converges. (b) It converges conditionally; the alternating series test guarantees convergence, but ∑∣ln⁡n∣n\displaystyle\sum \dfrac{|\ln n|}{n} diverges. (c) It diverges because ln⁡nn\dfrac{\ln n}{n} is not eventually decreasing, so the alternating series test does not apply. (d) It diverges because lim⁡n→∞ln⁡nn≠0\displaystyle\lim_{n\to\infty}\dfrac{\ln n}{n}\neq 0.

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