(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Does it converge, and what is the sum:

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

2. Relevant equations

3. 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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Series convergence, root test

**Physics Forums | Science Articles, Homework Help, Discussion**