How to evaluate this limit $\sum_{n=1}^\infty\frac{1}{n\sqrt[n]{n}}$ and its convergence?
I tried ratio test, root test, Raabe's test. However, I'm not getting anywhere. Can you please help me? Thank you
How to evaluate this limit $\sum_{n=1}^\infty\frac{1}{n\sqrt[n]{n}}$ and its convergence?
I tried ratio test, root test, Raabe's test. However, I'm not getting anywhere. Can you please help me? Thank you
For $n$ sufficiently large, $\root n\of n<2$; so, ${1\over n\,\root n\of n}>{1\over 2n}$ for sufficiently large $n$.
Since the series $\sum\limits_{n=1}^\infty {1\over 2n}$ diverges (it is essentially the harmonic series), it follows from the Comparison test that the series $\sum\limits_{n=1}^\infty {1\over n\,\root n\of n}$ diverges.
Hint: search an equivalent of $\sqrt[n]{n}$ as $n$ goes to $+\infty$ and use the Limit comparison test.