TITLE: Convergence of series $\sum\limits_{n=1}^\infty\int\limits_{1}^{+\infty}e^{-x^n}\,dx$
QUESTION [6 upvotes]: Determine if the series $\sum\limits_{n=1}^\infty \alpha_n$ converge, where
$$\alpha_n=\int\limits_{1}^{+\infty}e^{-x^n}\,dx.$$

Attempt. Ι am pretty sure the inequalities $$e^{-x^n}\leq \frac{1}{1+x^n}$$
and $e^{-x^n}\geq 1-x^n$ will be useful (the first one I believe more, which gives convergence for the series though, which I am not sure if it is correct).
Thanks for the help.

REPLY [3 votes]: $$\alpha_n =\frac{1}{n}\int_{1}^{+\infty} \frac{z^{1/n}}{z} e^{-z}=\frac{1}{en}\int_{0}^{+\infty}(z+1)^{1/n}\frac{dz}{e^z(z+1)} $$
is not a summable term since the dominated convergence theorem ensures 
$$ \lim_{n\to +\infty}\int_{0}^{+\infty}(z+1)^{1/n}\frac{dz}{e^z(z+1)}=\int_{0}^{+\infty}\frac{dz}{e^z(z+1)}\approx\frac{31}{52} $$
and the harmonic series is divergent.