Chapter 8.2, Problem 25E

To determine

To find: The series is convergent or divergent and find its sum if it is convergent.

Answer to Problem 25E

The series is divergent.

Explanation of Solution

Given information: The series is ${\displaystyle \sum _{n=1}^{\infty }\sqrt[n]{2}}$

Theorem: The necessary condition for convergence of the series ${\displaystyle \sum u_n}$ is that $\underset{x\to \infty }{\lim }{u}_n\to 0\text{ as }n\to \infty .$

Given, ${\displaystyle \sum _{n=1}^{\infty }\sqrt[n]{2}}$

Let, $u_k=\sqrt[n]{2}$

  $\Rightarrow u_k=\sqrt[n]{2}$

Take limit both side, we have

  $\Rightarrow \underset{x\to \infty }{\lim }{u}_k=\underset{x\to \infty }{\lim }\sqrt[n]{2}$

  $\Rightarrow \underset{x\to \infty }{\lim }{u}_k=1\ne 0$

From the above theorem the given series is divergent.