Chapter 8.1, Problem 27E

To determine

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

Answer to Problem 27E

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

Explanation of Solution

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

Solution:

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

  $\begin{array}{l}\underset{n\to \infty }{\lim }{s}_n=\underset{n\to \infty }{\lim }{\left(1+\frac{2}{n}\right)}^n\\ \Rightarrow \underset{n\to \infty }{\lim }{s}_n=e^2\left[\because \underset{n\to \infty }{\lim }{\left(1+\frac{a}{x}\right)}^x\to e^a\right]\end{array}$

  $\therefore$ Limit exists and is finite.

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