Problem with Series.

1. Nov 18, 2007

azatkgz

1. The problem statement, all variables and given/known data

Determine whether the series converges or diverges.

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

3. 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.

2. Nov 18, 2007

morphism

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

3. Nov 18, 2007

azatkgz

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

so both of them diverge

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