How to get linear or polynomial approximation of $\frac{1}{1-e^{-x}}$?

Question

In mathematical program, one of the constraint has $\frac{1}{1-e^{-x}}$ term which can't be solved by the linear/quadratic solver.

Can someone please provide a way to represent this term as a linear or quadratic approximation?

Thank you.

Answer 1

$$y=\frac{1}{1-e^{-x}}$$

Obviously a linear or quadratic approximation will be very bad. A main term $\frac{1}{x}$ is necessary.

We can use the series expansion of the function : $$y(x)\simeq \frac{1}{x}+\frac12+\frac{1}{12}x-\frac{1}{720}x^3+\frac{1}{30240}x^5-\frac{1}{1209600}x^7+....$$ But there is no need for so many terms for the approximate on the limited range $0<x\leq1$.

Below, the comparison of results for different numbers of terms of the series :