TITLE: How to determine the convergence of $\sum\limits_{n=1}^{\infty}\frac{n^{n^2}}{(n+x)^{n^2}}$
QUESTION [5 upvotes]: I've been studying for my analysis exams, and I've come across a series I haven't been able to solve. The question is just to determine for which real values of $x$ does the series converge. I've figured out it does converge for positive values and diverges for negative values of $x$. 
$$\sum_{n=1}^{\infty}\frac{n^{n^2}}{(n+x)^{n^2}}$$
However, the only value that I've actually been able to prove properly is $x=0$, and just because in that case it's trivially easy as the limit of the general term is $1$ (and hence it's not $0$).
Not sure how to prove the other cases. I'm guessing the way is probably factorizing the denominator, but I haven't gotten farther than that.

REPLY [5 votes]: Root Test:
$$\lim_{n\to\infty} |a_n|^{\frac{1}{n}} = \lim_{n\to\infty} |\frac{n^{n^2}}{(n+x)^{n^2}}|^{\frac{1}{n}} = \lim_{n\to\infty} \frac{n^n}{(n+x)^n} = \lim_{n\to\infty} \left( \frac{n}{n+x} \right)^n =...$$

REPLY [3 votes]: Hint: Use the Root Test. Note that 
$$\lim_{n\to\infty}\left(\frac{n+x}{n}\right)^n=e^x.$$