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## Homework Statement

Does it converge, and what is the sum:

[tex] \sum_{n=1}^{\infty}\frac{1}{n n^{\frac{1}{n}}} [/tex]

## Homework Equations

## The Attempt at a Solution

Please check my method and conclusion:

Using the root test:

[tex] \displaystyle\lim_{n\to\infty}\left|\frac{1}{n n^{\frac{1}{n}}} \right|^{\frac{1}{n}} = \displaystyle\lim_{n\to\infty}\frac{1}{n n^{\frac{1}{n}}} = \displaystyle\lim_{n\to\infty}\frac{1}{n}\displaystyle\lim_{n\to\infty}\frac{1}{ n^{\frac{1}{n}}} = 0 [/tex]

so while this doe not give me the sum (I'm not excited about that part...), it at least tells me that this series converges. Am I correct?