The exponential function can be defined via: $$ e^x = \lim_{n \rightarrow \infty} \left( 1 + \frac{x}{n} \right)^{n} = \lim_{n \rightarrow \infty} g(x; n) $$
In my problem, I actually have the right hand side $g(x;n)$, for large ($O(100)$), but finite $n$, which I need to evaluate numerically.
Are there ways to derive the corrections due to finite $n$; i.e. an asymptotic expansion in $n$?
That way I can evaluate something like $g(x;n) \approx e^{x} c(x;n)$ pretty easily, assuming $c(x;n)$ is a polynomial or rational function or otherwise easy to evaluate.