How does one work backwards from an infinite series to a function? How does one go about working backwards from an infinite series to a function?
For example, how would one determine what function this infinite series represents:
$$C + \frac{C}{x} + \frac{C}{x^2} + \frac{C}{x^3} +\cdots$$
where $C$ is any constant.

It depends very much on the series. Your example, however, isn’t too hard, because it’s geometric:
$$\sum_{n\ge 0}\frac{C}{x^n}=C\sum_{n\ge 0}\left(\frac1x\right)^n=C\frac1{1-\frac1x}=\frac{Cx}{x-1}\;,$$
and it converges for $\left|\frac1x\right|<1$, i.e., for $|x|>1$.
