Is there an exponential bound for $(1+p)^{-n}$ when $p$ is small? Is there an exponential upper bound for $(1+p)^{-n}$ when $0<p<1$? Similarly a exponential lower bound for $(1+p)^n$ will also be good.  Do you know of any resources where one can pick up bounds as such quickly? Thanks.

When $np \ll 1, (1\pm p)^n \approx 1 \pm np$ with the accuracy improving as $np$ gets smaller.  The error is of order $\frac {(np)^2}2$.  When you say exponential bound, do you want less than exponential growth?  It certainly doesn't fall exponentially
