calculus - Problem solving a series with convergence test: $sum_{k=1}^{infty}frac{k!}{(2k)!}$

Good morning, I have a big problem solving this:
$\sum_{k=1}^{\infty}\frac{k!}{(2k)!}\:$

I'm trying solving this limit with test of D'Alembert, but I have a problem solving the limit.

$\lim_{k\rightarrow\infty}\frac{(k+1)!k!}{(2k+2)!2k!}=(?)$

please, help me.

Answer

We have that
$$
\lim_{k\to\infty}\frac{(k+1)!(2k)!}{k!(2(k+1))!}=\lim_{k\to\infty}\frac{(k+1)(2k)!}{(2k+2)!}=\lim_{k\to\infty}\frac{k+1}{(2k+2)(2k+1)}=\lim_{k\to\infty}\frac1{2(2k+1)}=0.
$$
Hence, the series converges by the ratio test.