For what values of $a$, does the series $\sum^\infty_{n=1} {(n+a)^n \over {n^{n+a}}}$ converge? For what values of $a$, does the series $\sum^\infty_{n=1} {(n+a)^n \over {n^{n+a}}}$ converge?
I see that if I write $$\sum^\infty_{n=1} {(n+a)^n \over {n^{n+a}}} = \sum^\infty_{n=1} \left(1+ {a \over n}\right)^n {1 \over {n^a}}$$ then $a$ must be $a>0$ for $(1+ {a \over n})^n {1 \over {n^a}} \to 0$. (if $a \le 0$ then there is no convergence of course).
And if I compare with $1 \over {n^a}$ then the series converges if $a>1$.
But is that all?

The series$$\sum^\infty_{n=1} \left(1+ {a \over n}\right)^n {1 \over {n^a}}$$ converges for $a>1$. Indeed  $$ \left(1+ {a \over n}\right)^n {1 \over {n^a}}\sim_{\infty}\frac{e^a}{n^a}$$ that converges for $a>1$
