Is the series $\sum_{n=1}^{\infty} \Bigl(1-\Bigl(1-\frac{1}{n^{1+\epsilon}}\Bigr)^n\Bigr)$ convergent? While i was solving a problem in probability theory I came across the following series 
$$\sum_{n=1}^{\infty} \biggl(1-\biggl(1-\frac{1}{n^{1+\epsilon}}\biggr)^n\biggr)$$
and in order to complete my solution I want to show the above series converges but I couldn't prove it (I still dont know if it converges).
I tried to go with Taylor expansion of $x \mapsto \ln(1-x^{1+\epsilon})$ but I couldn't get anything interesting.
Then I showed that $1-\bigl(1-\frac{1}{n^{1+\epsilon}}\bigr)^n \leq \frac{1}{n^\epsilon}$ but this doesn't help either.
Let me know if you have any idea!

For large $n$, this term is approximately $n^{-\epsilon}$. The convergence condition is $\epsilon>1$.
