Test for convergence the series $$\sum_{n=1}^{\infty}\frac{1}{n^{(n+1)/n}}$$ I'd like to make up a collection with solutions for this series, and any new
solution will be rewarded with upvotes. Here is what I have at the moment
Method 1
We know that for all positive integers $n$, $n<2^n$, and this yields $$n^{(1/n)}<2$$ $$n^{(1+1/n)}<2n$$ Then, it turns out that $$\frac{1}{2} \sum_{n=1}^{\infty}\frac{1}{n} \rightarrow \infty \le\sum_{n=1}^{\infty}\frac{1}{n^{(n+1)/n}}$$ Hence $$\sum_{n=1}^{\infty}\frac{1}{n^{(n+1)/n}}\rightarrow \infty$$ EDIT:
Method 2
If we consider the maximum of $f(x)=x^{(1/x)}$ reached for $x=e$
and denote it by $c$, then $$\sum_{n=1}^{\infty}\frac{1}{c \cdot n} \rightarrow \infty \le\sum_{n=1}^{\infty}\frac{1}{n^{(n+1)/n}}$$
Thanks!