<s>[INST] Convergence of series with ^(n+1) 
I want to check, if this series is convergent or not: 
$$\sum\limits_{n=1}^\infty  \frac{n^n(n!)}{(2n)!}$$ 
I tried with the ratio test:
$$\frac{\frac{(n+1)^{n+1}(n+1)!}{(2n+2)!}}{\frac{n^n*n!}{(2n)!} } =\frac{(n+1)^{n+1}(n+1)!(2n)!}{(2n+2)!(n!)n^n}=\frac{(n+1)^{n+1}(n+1)}{n^n(2n+2)(2n+1)}$$
But how should I proceed? 
Thank you for your help

queenD
 [/INST] \begin{align}
\dfrac{n^n n!}{(2n)!} &= \dfrac{n^n}{(n+1)(n+2) \cdots 2n} \\
&= \prod_{k=1}^n \dfrac{n}{n+k} \leq \prod_{k= \lfloor n/2\rfloor + 1}^n \dfrac{n}{n+k} \\
&\leq \left(\dfrac{n}{n +  n/2}\right)^{n- \lfloor n/2\rfloor } = \left(\frac{2}{3}\right)^{n- \lfloor n/2\rfloor }\\ &\leq \left(\dfrac{2}{3}\right)^{n - n/2}=\left(\sqrt{\dfrac{2}{3}}\right)^n
\end{align}
</s>