Does the Series \(\sum_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}\) Converge?

[steviet]

$$\sum_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}\]$$

Does this series converge?

 

[Gib Z]

No it doesn't, because $$\sqrt[n]{n}}$$ approaches 1 as n approaches infinity. Hence if we were to the the first k partial sums of the series in your post, and the harmonic series, the difference would approach a constant as k approached infinity. Since the harmonic series diverges, so does this one.