Chapter 10.1, Problem 15E

To determine

Check whether the series converges or diverges. If converges, then find sum.

Answer to Problem 15E

The series diverges.

Explanation of Solution

Given Information:

The series is ${\displaystyle \sum _{n=0}^{\infty }\cos \left(n\pi \right)}$ .

Formula Used:

The sum of the geometric series when ration is less than 1:

  $S=\frac{a}{1-r}$ , where $a$ is initial term and $r$ is the common ratio.

Calculation:

The series is ${\displaystyle \sum _{n=0}^{\infty }\cos \left(n\pi \right)}$ .

Simplify the series.

  $\begin{array}{c}{\displaystyle \sum _{n=0}^{\infty }\cos \left(n\pi \right)}=\cos \left(0\right)+\cos \left(\pi \right)+\cos \left(2\pi \right)+\cos \left(3\pi \right)+\cdots \\ =1-1+1-1+\cdots \end{array}$

In the series ${\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n}$ , terms do not approach to 0.

Hence, the series diverges.