I wonder if there is Laurent expansion of $f(x) = 1 - e^{-x}$ where $x>0$, is real number.
It is known that the expansion : $$ g(x) = 1- e^{-x} = x - \frac{x^2}{2!} + \frac{x^3}{3!} - ... $$ for $ - \infty < x < \infty $
Originally function $f(x) \in (0,1)$ is bounded in $(0, \infty)$, however, the sum of the initial segment of $g(x)$ seems oscillatory too much as $x$ goes up, against that $f'(x) = e^{-x}$ trends to $0$ (I know that it may strictly make little sense, Anyway). So it makes me think whether or not there exist a Laurent expansion of $f(x)$ in any chance.
This is to find that $$ f(x) = c + \sum_{i=1} ^{\infty} a_i (x-w_0)^ {-i} + b_i (x-w_0)^i $$ where all of $a_i$ are not $0$, $c,w_0,b_i$ is real.