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