Composition of Power Series
Now that we can take powers of functions defined by power series and define them by power series in the same radii.. well, we’re all set to compose functions defined by power series!
Let’s say we have two power series expansions about :
within the radius , and
within the radius .
Now let’s take a with and . Then we have a power series expansion for the composite:
The coefficients are defined as follows: first, define to be the coefficient of in the expansion of , then we set
To show this, first note that the hypothesis on assures that , so we can write
If we are allowed to exchange the order of summation, then formally the result follows. To justify this (at least as well as we’ve been justifying such rearrangements recently) we need to show that
converges. But remember that each of the coefficients is itself a finite sum, so we find
On the other hand, in parallel with our computation last time we find that
So we find
which must then converge.