Method for increasing the bounds of a series

Question

My textbook claims that $$\sum_{n=0}^\infty a_n = x + \sum_{n=1}^\infty a_n$$

where $a_n$ is a Maclaurin series that is a function of $x$. How is this proved? This is most likely very simple, but I can't seem to figure it out.

Answer 1

Either $a_0=x,$ or the sum is infinite and $x$ is finite.