Power series

Let be a power series.

Radius of convergence

Each power series has a radius of convergence s.t. the series converges for all and diverges for all .

Proof sketch

Use something like ratio test i think.

Uniform convergence of power series

Let be the radius of convergence of a power series. Then for any the series Converges Uniformly on .

Proof sketch

Take s.t. , By convergence of power series find s.t. for all . Then we have , so

So take and note that converges (as a geometric series). Thus for any find such that for all . Then

for any . We are done by General Principle of Uniform Convergence.

Differentiation of power series

Suppose we have a powerseries with radius of convergence . Then the “derived series” also has radius of convergence (just do some ineqs) Now the original powerseires has a point where it converges and it’s derivative converges uniformly for every . Let . Pick . Then using convergence of derivatives we find the derivative of the powerseries at is exactly