Convergent by Ratio test? I am lost with this problem: $$\sum_{n=1}^\infty \frac{n^n}{2^n n!}.$$
I am suppose to find if it is convergent or divergent.
I have the correct set up. After cancelling everything I am left with this:
$$\lim \frac{ (n+1)^{n+1} }{ 2(n+1)n^n }$$
How do I get the limit from that?

We have $a_n=\frac{n^n}{2^n\,n!}$.  Thus, 
$$\frac{a_{n+1}}{a_n}=\frac{(n+1)^{n+1}}{2^{n+1}(n+1)!}\frac{2^n\,n!}{n^n}=\frac{1}{2}\left(1+\frac1n\right)^n\to \frac{e}{2}>1\implies \text{the series diverges}$$
