Differentiating power series

Gowers's Weblog

I’m writing this post as a way of preparing for a lecture. I want to discuss the result that a power series $latex sum_{n=0}^infty a_nz^n$ is differentiable inside its circle of convergence, and the derivative is given by the obvious formula $latex sum_{n=1}^infty na_nz^{n-1}$. In other words, inside the circle of convergence we can think of a power series as like a polynomial of degree $latex infty$ for the purposes of differentiation.

A preliminary question about this is why it is not more or less obvious. After all, writing $latex f(z)=sum_{n=0}^infty a_nz^n$, we have the following facts.

  1. Writing $latex S_N(z)=sum_{n=0}^Na_nz^n$, we have that $latex S_N(z)to f(z)$.
  2. For each $latex N$, $latex S_N'(z)=sum_{n=1}^Nna_nz^{n-1}$.

If we knew that $latex S_N'(z)to f'(z)$, then we would be done.

Ah, you might be thinking, how do we know that the sequence $latex (S_N'(z))$ converges? But it turns out that that is not the problem: it…

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