Chapter 10.5, Problem 13E

To determine

To find: that the whether the series converges or diverges.

Answer to Problem 13E

Diverges

Explanation of Solution

Given:

  ${\displaystyle \sum _{n=1}^{\infty }n\sin \left\{\frac{1}{n}\right\}}$

Calculation:

  $\begin{array}{l}{\displaystyle \sum _{n=1}^{\infty }{a}_n={\displaystyle \sum _{n=1}^{\infty }n\sin \left\{\frac{1}{n}\right\}}}\\ a_n=n\sin \left\{\frac{1}{n}\right\}\\ \underset{n\to \infty }{\lim }n\sin \left\{\frac{1}{n}\right\}\\ \underset{n\to \infty }{\lim }\frac{\sin \left\{\frac{1}{n}\right\}}{\frac{1}{n}\cdot }\end{array}$

Replace $\frac{1}{n}$ with t, To get

  $\underset{t\to 0}{\lim }\frac{\sin \left\{t\right\}}{t}=1$

Hence, by limit test the series is divergent