While reading the Wikipedia article on the Binomial transform, which is defined for a given sequence $\{a_n\}$ as $$ s_n = \sum_{k=0}^{n}(-1)^k \binom{n}{k} a_k $$ I found the following relation between generating functions: If $f(x) = \sum_{n \ge 0} a_n x^n$ and $g(x) = \sum_{n \ge 0} s_n x^n$ then $$ (1-x) g(x) = f\left( \frac{x}{x-1}\right) $$ which recalling the original definitions for $g(x),f(x)$ and $s_n$ can be rewritten as $$ \sum_{n\ge 0} a_n x^n = \sum_{n\ge 0} \left(\sum_{k=0}^{n}(-1)^{k+1} \binom{n}{k}a_k\right) \frac{x^n}{(x-1)^{n+1}} $$
The Wikipedia article doesn't mention how to prove this relationship between the generating functions, and I wondered how to prove it. My first idea was to expand $(x-1)^{-n-1}$ as a Taylor series such that we have several sums, but where only terms of the form $x^{\text{Something}}$ is present which resembles the original form we want to get at. This didn't seem to lead anywhere. Does anyone have any resources or suggestions on how to prove this? Thanks!