Mathematics

Problem

Consider the series

\displaystyle \sum_{n=2}^{\infty} (-1)^{n}\dfrac{\ln n}{n}

. Which statement best describes its convergence? a) It converges absolutely; for large

n

\dfrac{\ln n}{n}<\dfrac{1}{n^{1.1}}

, and the

p

-series with

p=1.1>1

converges. b) It converges conditionally; the alternating series test guarantees convergence, but

\displaystyle\sum \dfrac{|\ln n|}{n}

diverges. c) It diverges because

\dfrac{\ln n}{n}

is not eventually decreasing, so the alternating series test does not apply. d) It diverges because

\displaystyle\lim_{n\to\infty}\dfrac{\ln n}{n}\neq 0