Mathematics

Mathematics

Problem

Consider the series ∑n=5∞(−1)nln⁡nn2\displaystyle \sum_{n=5}^{\infty} (-1)^{n}\dfrac{\ln n}{n^{2}}. Which statement best justifies whether the series converges? a) It converges by the Ratio Test since lim⁡n→∞∣an+1an∣=0<1\displaystyle\lim_{n\to\infty}\left|\tfrac{a_{n+1}}{a_n}\right|=0<1. b) It diverges because ln⁡n\ln n grows without bound, so the terms of the series fail to approach 00. c) It converges because (−1)nln⁡nn2(-1)^{n}\dfrac{\ln n}{n^{2}} satisfies the alternating series test: ln⁡nn2\dfrac{\ln n}{n^{2}} is positive, eventually decreasing, and approaches 00 as n→∞n\to\infty. d) It converges conditionally but not absolutely, because the alternating series test is satisfied while ∑∣ln⁡n∣n2\displaystyle\sum\frac{|\ln n|}{n^{2}} behaves like ∑ln⁡nn\sum\frac{\ln n}{n}, which diverges.

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