Mathematics

Mathematics

Problem

Consider the series ∑n=2∞ln⁡nn\displaystyle\sum_{n=2}^{\infty} \frac{\ln n}{n}. Which statement best justifies its divergence? a) Because the ratio test gives lim⁡n→∞an+1an=0<1\displaystyle\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=0<1, the series converges absolutely by the ratio test. b) By the limit comparison test with 1n\displaystyle\frac1n, the limit is a finite number, so the series converges. c) The limit of the ratio an+1an\dfrac{a_{n+1}}{a_n} equals 11, so the ratio test is inconclusive; evaluating ∫2∞ln⁡xx dx\displaystyle\int_{2}^{\infty} \frac{\ln x}{x}\,dx shows the series diverges. d) Evaluating ∫2∞ln⁡xx dx=12(ln⁡2)2\displaystyle\int_{2}^{\infty} \frac{\ln x}{x}\,dx=\tfrac12(\ln 2)^2 (a finite value) shows the series converges by the integral test.

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