Power Series
Definition (Power Series)
Let be a complex Banach space. A power series (with values in ) is an infinite series of the form
where is the th coefficient, is a complex variable and is the center of the series.
Suppose the Banach space valued power series converges for . Then it converges absolutely for all with .
Proof TODO
Suppose is a Banach space valued power series. Then either
- the series converges only for (formally ), or
- there exists a number such that the series converges absolutely whenever and diverges whenever , or
- the series converges absolutely for all (formally ).
The number is called the radius of convergence of the power series.
Proof TODO
Cauchy–Hadamard Formula
Let be a Banach space valued power series with radius of convergence . Then
Proof TODO