approximations of finite series of $e^x$

Question

Are there any common identies or rules for approximating finite summations of $e^x$? Specifically looking at why $\frac{2\delta}{e^\delta}\sum_{i=0}^{n-1}e^{-2\delta i} \approx 1 - e^{-2 \delta n}$ for $\delta \ll 1$

Answer 1

This is just the geometric progression: $$\sum_{i=0}^N r^i = \frac{1-r^{N+1}}{1-r}$$ with $r = e^{-2\delta}$ and $N=n-1$. Note that $$\frac{2\delta}{e^\delta} \sum_{i=0}^{n-1} (e^{-2\delta})^i = \frac{2\delta}{e^\delta} \frac{1-(e^{-2\delta})^n}{1-e^{-2\delta}}$$ now use the fact that $\delta < 1$.