I'm reading Fan Chung's Spectral Graph Theory. There's a pair of inequalities I don't know how to justify. Chung doesn't attempt to explain them, so maybe they're very obvious. Example 1.19 on page 20 reads, in part:
$ \frac{1}{2} \bigg( \sum_{k=1}^n \binom{n}{k} (1 - \frac{2k}{n+1})^{2s} \bigg)^{1/2} \leq \frac{1}{2} \bigg( \sum_{k=1}^n e^{k\log n - \frac{4ks}{n+1}} \bigg)^{1/2} \leq e^{-c} $ if $s \geq \frac{1}{4}n\log n +cn$
($n$ and $s$ are arbitrary positive integers, representing the number of vertices in a graph and the number of steps in a random walk, respectively.)
The first inequality seems to suggest that
$ \binom{n}{k}(1-\frac{2k}{n+1})^{2s} \leq e^{k\log n - \frac{4ks}{n+1}} $
I do know that $(1 + x/n)^n \approx e^x$ for large $n$, and there's a faint resemblance to the inequality above. Could that be part of the answer?
(An excerpt of the book which contains the part I'm talking about is available here: http://www.math.ucsd.edu/~fan/research/cbms.pdf )