Suppose I have an analytic function $f$ defined by some Taylor series around $x=0$, such that $$f(x) = \sum_{n=1}^\infty a_n x^n. $$
I now wish to consider the Taylor expansion of $$(1+f(x))^{1/x} = \sum_{m=0}^\infty b_m x^m, $$ and would like to express the coefficients $b_m$ in terms of $a_n$.
A first attempt using binomial expansion gives: $$ (1+f(x))^{1/x} = 1 + \frac{1}{x}f(x) + \frac{1}{2x}(\frac{1}{x}-1)f^2(x)+ \frac{1}{3!x}(\frac{1}{x}-1)(\frac{1}{x}-2)f^3(x) \dots $$ but it's not clear to me what the $b_m$ is from this.