An important but slightly technical result on power series states that if the power series has radius of convergence then its derivative is , where this series has radius of convergence at least . An interesting post on Gowers’ Weblog gives a direct proof of this.
The main purpose of this post is to record a slightly different direct proof. As in Gowers’ proof we need to know that has radius of convergence at least . For completeness here is the standard proof: replace with where and observe that converges absolutely, while is bounded.
Now fix such that and choose such that . Since and both converges absolutely and uniformly on the closed disc of radius there exists such that, whenever , the errors in approximating
- by ,
- by ,
are all . Since
for some (complicated) polynomial , the left-hand side is for all sufficiently small . Hence
for all sufficiently small . The result follows.
I like this proof because all the heavy work is done by the absolute and uniform convergence of power series with their circle of convergence. Okay, this takes some work to prove, but the trick used above to show that has radius of convergence at least can be applied, or one can use the Weierstrass -test.
There are several alternatives that are probably more conceptual, but depend on more technology. For example, there is a real variable proof which starts by showing that converges uniformly on a closed disc inside the circle of convergence, and then uses integration (of a uniform limit of continuous functions) to show that has antiderivative .
Here is an outline of a complex variable proof. Once all the ingredients are in place, it takes little work, but the taste is probably a bit artificial. Let and let denote the closed unit disc of radius in . Let . Then uniformly on . As a uniform limit of holomorphic functions, is holomorphic (Morera's Theorem). By the derivative version of Cauchy's Integral Formula,
where is a circular contour about contained in . Since converges uniformly to on this contour as , we have
and the result follows.