Chapter 9.5, Problem 31E

To determine

To calculate: The series converges absolutely, converges conditionally, or diverges.

Answer to Problem 31E

The series converges conditionally.

Explanation of Solution

Given information:

  ${\displaystyle \sum _{n=1}^{\infty }\frac{\cos (n\pi )}{n}}$

Calculation:

Assuming it is intended to be like how wrote it with parentheses here (and not $(\cos n)$ times $\pi$ ), the function is just like ${(-1)}^n\frac{1}{n}$

  $\begin{array}{c}{\displaystyle \sum _{n=1}^{\infty }\frac{\cos (n\pi )}{n}}=-\frac{1}{1}+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\dots \\ \underset{n\to \infty }{\lim }\frac{1}{n}=0\end{array}$

Alternating series test

  $\frac{1}{n}$ is positive.

The limit of the terms approaches zero.

Each successive term (without the sign) is smaller than the previous term.

So it converges.

(probably useless info: this series $=-\ln 2$ )

To test for absolute convergence, can drop the $\cos (n\pi )$ part, as it acts as the alternating $+1,-1$ .

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

  $\frac{1}{n}$ is know to diverge. So it doesn't converge absolutely.

Therefore, the original series converges and it doesn't converge absolutely so it conditionally converges.