Chapter 8.1, Problem 24E

To determine

To find: To determine whether the sequence converges or diverges and find limit of converges.

Answer to Problem 24E

The series ${\displaystyle \sum _{n=0}^{\infty }{a}_n}$ is converges and its value is ${\displaystyle \sum _{n=0}^{\infty }{a}_n=\frac{\pi }{2}}$

Explanation of Solution

Given information: $a_n=\arctan 2n$

Solution:

To determine converges or diverges we need to check the limit of the sequence of the partial sums.

  $\begin{array}{l}\underset{n\to \infty }{\lim }{s}_n=\underset{n\to \infty }{\lim }\arctan 2n\\ \Rightarrow \underset{n\to \infty }{\lim }{s}_n=\frac{\pi }{2}\\ \because {\tan }^{-1}\left(\infty \right)=\frac{\pi }{2}\end{array}$

  $\therefore$ Limit exists and is finite.

Hence, The series ${\displaystyle \sum _{n=0}^{\infty }{a}_n}$ is converges and its value is ${\displaystyle \sum _{n=0}^{\infty }{a}_n=\frac{\pi }{2}}$