combinatorics - Upper bound on sum of binomial coefficients

If have been trying to prove the following limit:

$\displaystyle\lim_{n\to \infty}\dfrac{\binom{n}{n/2}+\cdots+\binom{n}{n/2+\sqrt{n}}}{2^n}=0$
using the chernoff bounds for $\displaystyle\sum_{i=0}^{n/2+\sqrt{n}}\binom{n}{i}$ and then
I suppose that the sum first $n/2$ terms should be easy to calculate and subtract.
But thus far I have had no success in proving this.

I've thought of using Stirling's formula for each binomial coefficient,
but then I see that that may not lead to anything.

I would appreciate I you could give me a hint to start my proof.