Upper bound for $S_{n,r}=\sum_{i=0}^n \binom{n}{i} \left(\frac{\binom{n}{i}}{2^n}\right)^{r}$.

Question

I am trying to get an upper bound for $$S_{n,r}=\sum_{i=0}^n \binom{n}{i} \left(\frac{\binom{n}{i}}{2^n}\right)^{r} .$$

Any hints would be greatly appreciated. I thought of using Stirling's approximation but I worried that doesn't give good bounds for binomial coefficients over the full range.

Answer 1

Actually, you can do a lot better than @Xoff, since $$\binom{n}{i} \leq 2^n/\sqrt{n}$$, so $$S_{n, r} \leq 2^n/n^{r/2}.$$

Answer 2

As $\binom{n}{i}\le 2^n$, $$S_{n,r}\le\sum_{i=0}^n\binom{n}{i}.1^r=2^n$$

if you want something better you can use $\binom{n}{i}\le\binom{n}{p}$ (with $p=\lfloor\frac{n}{2}\rfloor$).

Hence $$S_{n,r}\le 2^n\left(\frac{\binom{n}{p}}{2^n}\right)^r$$

So $\lim_{r\rightarrow\infty}S_{n,r}=0$.