TITLE: Convergence of $\sum\limits_{n = 1}^{+\infty} e^{-\sqrt{n + 1}}$
QUESTION [3 upvotes]: I need to show, using the comparison test, that $\sum\limits_{n = 1}^{+\infty} e^{-\sqrt{n + 1}}$ converges, but I can't come up with a larger convergent series.
Thanks.

REPLY [7 votes]: Since $\mathrm e^u=\sum\limits_{k=0}^{+\infty}\frac{u^k}{k!}\gt\frac{u^4}{4!}$ for every $u\gt0$, one has $\mathrm e^{-\sqrt{n+1}}\lt\frac{4!}{(n+1)^2}$ hence the series converges.
A similar argument shows that the series $\sum\limits_{n}\mathrm e^{-n^a}$ converges for every $a\gt0$.
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