# Chapter 11 - Infinite Series - Chapter Review Exercises - Page 592: 87

converges

#### Work Step by Step

Given $$\sum_{n=1}^{\infty} \frac{\cos n}{n^{3 / 2}}$$ We check $\sum_{n=1}^{\infty} |\frac{\cos n}{n^{3 / 2}}|$ Since \begin{align*} |\cos n | &\leq 1\\ \frac{|\cos n |}{n^{3/2}} &\leq \frac{1}{n^{3/2}} \end{align*} We know that $\sum \frac{1}{n^{3/2}}$ is a convergent $p\gt1$ series. Thus, $\sum_{n=1}^{\infty} |\frac{\cos n}{n^{3 / 2}}|$ also converges and hence the given series converges.

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