Consider the function $$f(x) = \frac{x}{1-x}$$ We know that for $x\in(0,1)$, $$f(x) = x\cdot\frac1{1-x} = \sum_{k=0}^\infty x^{k+1} = \sum_{k=0}^\infty x^{k} - 1$$ Now, notice that: $$\frac{x}{1-x} = \frac{x^{-1}}{x^{-1}}\frac{x}{1-x} = \frac1{x^{-1}-1}$$ Which reveals that for $x\in(1,\infty)$ $$f(x) = -\sum_{k=0}^\infty x^{-k}$$ So we now have two different infinite sum representations for $f$ across $[0,\infty)\cap\{x\neq1\}$.
My question is as follows: Given a function $f$ which has a taylor series expansion for some interval, are there approaches or theorems which allow you to find other expansions which work on other intervals? There wasn't any magic in what I did above, I just happened to notice I could write the function in a slightly different way. Wondering if there are known approaches or relationships like this for other functions.
