Solving Series: Does \sum_{n=1}^{\infty}\frac{1}{n^{1+\frac{1}{n}}} Converge?

[azatkgz]

Homework Statement

Determine whether the series converges or diverges.


$$\sum_{n=1}^{\infty}\frac{1}{n^{1+\frac{1}{n}}}$$

The Attempt at a Solution

$$\sum_{n=1}^{\infty}\frac{1}{nn^{\frac{1}{n}}}=\sum_{n=1}^{\infty}\frac{1}{ne^{\frac{1}{n}\ln n}}$$

$$\lim_{n\rightarrow\infty}\frac{\ln n}{n}=0$$

$$\sum_{n=1}^{\infty}\frac{1}{ne^0}=\sum_{n=1}^{\infty}\frac{1}{n}$$

Series diverges.

 

[morphism]

Are you applying some form of the limit comparison test? If so, then you're right.

 

[azatkgz]

$$\lim_{n\rightarrow\infty}\frac{1/ne^{\frac{1}{n}\ln n}}{1/n}=1$$

so both of them diverge