Chapter 11 - Infinite Series - 11.3 Convergence of Series with Positive Terms - Exercises - Page 556: 37

Converges

Given $$\sum_{n=1}^{\infty}\sin \frac{1}{n^{2}} $$ Since for $n\geq1$ \begin{aligned} \sin \frac{1}{n^{2}} \leq \frac{1}{n^{2}} \end{aligned} Compare with $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{2}} $, a convergent $p-$series $p=2 $; then $\displaystyle\sum_{n=1}^{\infty}\sin \frac{1}{n^{2}} $ also converges.