Showing that $\lim\limits _{n\to\infty}{n \choose \left\lceil \frac{n}{2}\right\rceil }\cdot2^{-n}=0$ but the matching series does not converge

Question

I want to show that: $$\lim\limits _{n\to\infty}{n \choose \left\lceil \frac{n}{2}\right\rceil }\cdot2^{-n}=0 $$ And also $${\displaystyle \sum_{n=1}^{\infty}{n \choose \left\lceil \frac{n}{2}\right\rceil }\cdot2^{-n}} $$ Does not converge. I'm not particularly good with this sort of approximation so I would really appreciate some help.

Answer 1

These are the central binomial coefficients. You can see the wiki page for the estimates $$\frac{2^n}{n+1}\le {n\choose \lceil \frac{n}{2}\rceil} \le \frac{2^n}{\sqrt{1.5n+1}}$$

Multiply through by $2^{-n}$ and you get your terms. The upper bound gives the first part; the lower bound gives the second part by comparison with the harmonic series.