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