Formally, we defined the product of two power series to be the series you get when you multiply out all the terms and collect terms of the same degree. Specifically, consider the series and . Their product will be the series , where the coefficients are defined by
Now if the series converge within radii and , respectively, it wouldn’t make sense for the product of the functions to be anything but whatever this converges to. But in what sense is this the case?
Let’s take a point inside both of the radii of convergence. Then we know that the series and both converge absolutely. We want to consider the product of these limits
Since the limit of the first sequence converges, we’ll just take it as a constant and distribute it over the other:
And now we’ll just distribute each across the sum it appears with:
And now we’ll use the fact that all the series in sight converge absolutely to rearrange this sum, adding up all the terms of the same total degree at once, and pull out factors of :
As a special case, we can work out powers of power series. Say that within a radius of . Then within the same radius of we have
where the coefficients are defined by