I'm trying to prove that if

An <= (1 - 1/n^a)^n

(when 0 < a < 1)

Then the series ∑An converges.

I managed to get to the form of

∑ (1/e)^(n^b)

(where 0<b<1)

But I can't even prove that this more simple series converges.

Could someone help me please?

Thanks in advance!