Test Convergence of $\sum_{k=1}^{\infty}= \frac{(-1)^{k+1} k^{k} }{(k+1)^{k}}$ I am trying to test the above series for convergence. First of all, it does not satisfy the conditions for the Alternating Test or the Dirichlet Test, since:  $\lim_{k \to \infty}a_{k} \to 1/e$.
I know that the series diverges, but I am not exactly sure how to prove it. I've considered other tests but they don't seem to be applicable.
Some of my classmates have said that $\lim_{k \to \infty}a_{k} \neq 0 \Rightarrow \sum a_{k}$ diverges. I don't necessarily agree with them, but I haven't really found a counterexample. Any help/suggestions would be appreciated.

Your classmates are right. Suppose that $\sum a_n$ converges to $a$ and let $S_n$ be the sequence of partial sums. We have $S_n \to a$, hence $S_n - S_{n-1} \to a - a = 0$. But $S_n - S_{n-1} = a_n$. Hence $a_n \to 0$.
So if you proved that $\lim a_n$ does not exist or that it is non-zero, then you can conclude that the series diverges.
Now let $a_n$ be the general term of your series. We have $a_{2n -1} \to 1/e$ and $a_{2n} \to -1/e$. Hence $\lim a_n$ does not exist.
