Does $\sum_{n=1}^{\infty} \frac{1}{n^{1 + 1/n}}$ converge?
Does $$\sum_{n=1}^{\infty} \frac{1}{n^{1 + 1/n}}$$ converge? If yes, to where?
I searched this specific series but couldn't find a solution.
Hint: $\dfrac{1}{n^{1+1/n}} = \frac{1}{n} \exp(-\frac{1}{n}\ln(n)) > \frac{1}{n} \exp(-1)$ as $n \to \infty$.
$2^n\ge n$ for $n\ge1$ is easy to show. Hence $2\ge n^{1/n}$, so
$${1\over n^{1+1/n}}\ge{1\over2n}$$
which implies $\sum{1\over n^{1+1/n}}\ge{1\over2}\sum{1\over n}$, which diverges.