Does the following series diverge or converge.
$\sum\frac{(k!)^2}{2k!}$
I decide to apply the powerful ratio test.
I did
$\frac{(k+1)!^2}{2(k+1)!}$
which then becomes
$\frac{(k+1)!(k+1)!}{(2k+2)!}$*$\frac{2k!}{(k!)(k!)}$
I get the
$k\rightarrow\infty$
$\frac{(k+1)(k+1)}{(2k+2)(2k+2)}$
=$\frac{1}{4}$
$\frac{1}{4}$<1
convergent?
Best Answer
Assuming you're working with the series:
$$\sum\frac{(k!)^2}{(2k)!},$$ then you should be able to reduce the limit of the ratio to $$\lim_{k \to \infty} \frac{(k+1)(k+1)}{(2k+2)(2k+1)}$$
The limit is still $\frac 14$, and so, yes, by the ratio test, we have that the series converges.