How to find out if a sequence with exponentiation in fraction is convergent

Question

I need to find the convergence of this function: $\sum^{\infty}_{x=1}{\frac{(x+1)^{x^2}}{x^{x^2}2^x}}$

Now my problem is, I have no clue how to do this (I tried the root-test and it did not work out)...

I thought the direct-comparison-test might do the job, but I am somehow not able to use it here, because I don't know how.

Any help would be gratefully accepted... =)

Answer 1

Hint: Your series is

$$\sum_{n=1}^{\infty}\frac{(1+1/n)^{n^2}}{2^n} = \sum_{n=1}^{\infty}\frac{[(1+1/n)^n]^n}{2^n}.$$