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> Determine if the series $\sum\limits\_{n=1}^\infty \alpha\_n$ converge, where
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> $$\alpha\_n=\int\limits\_{1}^{+\infty}e^{-x^n}\,dx.$$
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*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.