I was studying Random Walks and derived these two series.
Let $p\in (0,1)\setminus \{\frac{1}{2}\}$, i.e. $p\in (0,\frac{1}{2})\cup (\frac{1}{2}, 1).$
I want to prove the following series is convergent, i.e.
$$\sum_{k=1}^\infty \frac{2k!}{k!k!} p^k(1-p)^k < \infty$$
Meanwhile, I also want to prove the following series is divergent, i.e. $$\sum_{k=1}^\infty \frac{2k!}{k!k!} \left( \frac{1}{2} \right)^{2k} = \infty$$
I have no idea of dealing with the fraction of factorials. What I have observed so far is just that, $p^k(1-p)^k < (\frac{1}{2})^{2k}\,$ for any $p\in (0,\frac{1}{2})\cup (\frac{1}{2}, 1),$ which makes sense of the divergence and convergence of the two series above.
Any help or hint will be appreciated.
Hint: Use Stirling's approximation: $$ n!\sim\sqrt{2\pi n}\left(\frac n e\right)^n $$ to arrive at (writing $q:=1-p$) $$ \frac{(2k)!}{k!k!}(pq)^k \sim \frac {(4pq)^k}{\sqrt{\pi k}} $$