Mathematics

Mathematics

Problem

Determine what the nnth-term test concludes about the series ∑n=1∞nn+1e−1/n\displaystyle\sum_{n=1}^{\infty}\frac{n}{n+1}e^{-1/n}. a) The series converges because e−1/n→0e^{-1/n}\to 0, so the terms of the series approach 00. b) The nnth-term test is inconclusive because lim⁡n→∞nn+1e−1/n=0\displaystyle\lim_{n\to\infty}\frac{n}{n+1}e^{-1/n}=0. c) The series diverges because lim⁡n→∞nn+1e−1/n=1≠0\displaystyle\lim_{n\to\infty}\frac{n}{n+1}e^{-1/n}=1\neq 0. d) The series converges by the ratio test since lim⁡n→∞an+1an=e−1<1\displaystyle\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=e^{-1}<1.

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