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