Chapter 8.3, Problem 19E

To determine

To prove: ${\displaystyle \sum _{n=1}^{\infty }\frac{{\cos }^{2}n}{n^2+1}}$ is convergent or not .

Answer to Problem 19E

  ${\displaystyle \sum _{n=1}^{\infty }\frac{{\cos }^{2}n}{n^2+1}}$ is convergent.

Explanation of Solution

Given information:

  ${\displaystyle \sum _{n=1}^{\infty }\frac{{\cos }^{2}n}{n^2+1}}$

Calculation:

  $\left|\frac{{\cos }^{2}n}{n^2+1}\right|\le \frac{1}{n^2}$ [since ${\cos }^{2}n<1$ ]

We know that ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2}}$ is covergent p-series with p=2>1

Hence , ${\displaystyle \sum _{n=1}^{\infty }\left|\frac{{\cos }^{2}n}{n^2+1}\right|}$ is convergent by comparison test .

Consequently , ${\displaystyle \sum _{n=1}^{\infty }\frac{{\cos }^{2}n}{n^2+1}}$ is an absolutely convergent series .[since , an absolutely convergent series is convergent]

Hence , ${\displaystyle \sum _{n=1}^{\infty }\frac{{\cos }^{2}n}{n^2+1}}$ is convergent.