# Differentiating power series

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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