So we know that we can have two power series expansions of the same function about different points. How are they related? An important step in this direction is given by the following theorem.
Suppose that the power series converges for , and that it represents the function in some open subset of this disk. Then for every point there is some open disk around of radius contained in , in which has a power series expansion
The proof is almost straightforward. We expand
Now we need to interchange the order of summation. Strictly speaking, we haven’t established a condition that will allow us to make this move. However, I hope you’ll find it plausible that if this double series converges absolutely, we can adjust the order of summations freely. Indeed, we’ve seen examples of other rearrangements that all go through as soon as the convergence is absolute.
Now we consider the absolute values
Where we set . But then , where the last inequality holds because the disk around of radius fits within , which fits within the disk of radius around . And so this series of absolute values must converge, and we’ll take it on faith for the moment (to be shored up when we attack double series more thoroughly) that we can now interchange the order of summations.
This result allows us to recenter our power series expansions, but it only assures that the resulting series will converge in a disk which is contained within the original disk of convergence, so we haven’t necessarily gotten anything new. Yet.