Chapter 8.1, Problem 17E

To determine

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

Answer to Problem 17E

The series ${\displaystyle \sum _{n=0}^{\infty }{a}_n}$ is converges and its value is ${\displaystyle \sum _{n=0}^{\infty }{a}_n=1}$

Explanation of Solution

Given information: $a_n=\tan \left(\frac{2n\pi }{1+8n}\right)$

Solution:

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

  $\begin{array}{l}\underset{n\to \infty }{\lim }{s}_n=\underset{n\to \infty }{\lim }\tan \left(\frac{2n\pi }{1+8x}\right)\\ \Rightarrow \underset{n\to \infty }{\lim }{s}_n=\underset{n\to \infty }{\lim }\left(\frac{\frac{2n\pi }{n}}{\frac{1+8n}{n}}\right)\\ \Rightarrow \underset{n\to \infty }{\lim }{s}_n=\underset{n\to \infty }{\lim }\left(\frac{2\pi }{\frac{1}{n}+8}\right)\\ \left[\because \underset{n\to \infty }{\lim }\frac{1}{n}=0\right]\\ \Rightarrow \underset{n\to \infty }{\lim }{s}_n=\tan \left(\frac{2\pi }{8}\right)\\ \Rightarrow \underset{n\to \infty }{\lim }{s}_n=\tan \left(\frac{\pi }{4}\right)\\ \Rightarrow \underset{n\to \infty }{\lim }{s}_n=1\end{array}$

  $\therefore$ Limit exists and is finite.

Hence, The series ${\displaystyle \sum _{n=0}^{\infty }{a}_n}$ is converges and its value is ${\displaystyle \sum _{n=0}^{\infty }{a}_n=1}$