Is $\sum_{t=1}^{\infty} e^{-\sqrt{t}}$ finite?

Is $\sum_{t=1}^{\infty} e^{-\sqrt{t}}$ finite?

Is $\displaystyle \sum_{t=1}^{\infty} e^{-\sqrt{t}}$ finite? This is not
homework. Actually, a computer algebra package has told me it is finite,
but I would like to see a proof. No approach worth trying has occurred to
me - a hint would be appreciated...