Determine if the series rⁿ |sin (nx)| converges or diverges

by RoRi

March 8, 2016

Determine if the following series converges or diverges and justify your decision.

$\sum_{n=1}^{\infty} r^n |\sin (nx)|, \qquad r> 0.$

First, if $0 < r < 1$ the series converges by the comparison test since

$r^n | \sin (nx)| \leq r^n$

which implies

$\sum_{n=1}^{\infty} r^n |\sin(nx)| \leq \sum_{n=1}^{\infty} r^n$

which we know converges for $0 < r < 1$ . If $r \geq 1$ then in order for the series to converge we must have $|\sin (nx)| = 0$ (otherwise $\lim_{n \to \infty} a_n \neq 0$ so the series cannot converge). Thus, the series also converges when $x = k \pi$ for $k \in \mathbb{Z}$ .