Chapter 10, Problem 52RE

To determine

To check: if the series converges absolutely, converges conditionally or diverges with reason.

Answer to Problem 52RE

The series will diverge since $\underset{n\to \infty }{\lim }{T}_n\ne 0$

Explanation of Solution

Given:

The given series is, ${\displaystyle \sum _{n=1}^{\infty }{\left(\frac{n}{n+1}\right)}^n}$

Calculation:

For the given series, the $n^{th}$ term is given as

  $T_n={\left(\frac{n}{n+1}\right)}^n$

So.

  $\begin{array}{l}\underset{n\to \infty }{\lim }{T}_n=\underset{n\to \infty }{\lim }{\left(\frac{n}{n+1}\right)}^n\\ =\underset{n\to \infty }{\lim }{\left(1+\frac{1}{n}\right)}^{-n}\\ =\frac{1}{e}\ne 0\end{array}$

As, $\underset{n\to \infty }{\lim }{T}_n\ne 0$

So, the series will diverge.