Mathematics

Mathematics

Problem

Let an=nsin⁡ ⁣(1n)a_n = n\sin\!\left(\tfrac{1}{n}\right) and consider the series ∑n=1∞an\displaystyle\sum_{n=1}^{\infty} a_n. Based on the limit of ana_n as n→∞n\to\infty, which statement is correct? a) The series converges because lim⁡n→∞an=0\displaystyle\lim_{n\to\infty} a_n = 0. b) The series diverges because lim⁡n→∞an=1\displaystyle\lim_{n\to\infty} a_n = 1. c) The test is inconclusive because lim⁡n→∞an=0\displaystyle\lim_{n\to\infty} a_n = 0. d) The series converges because lim⁡n→∞an=1\displaystyle\lim_{n\to\infty} a_n = 1.

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