Power Series

Definition (Power Series)

Let $X$ be a complex Banach space. A power series (with values in $X$) is an infinite series of the form

$$\sum _{n=0}^{\infty }{x}_n(z-a{)}^n=x_0+x_1(z-a)+x_2(z-a{)}^2+\cdots ,$$

where $x_n\in X$ is the $n$th coefficient, $z$ is a complex variable and $a$ is the center of the series.

Suppose the Banach space valued power series ${\sum }_{n=0}^{\infty }{x}_n(z-a{)}^n$ converges for $z=a+w$. Then it converges absolutely for all $z$ with $|z-a|<|w|$.

Proof   TODO $\square$

Suppose ${\sum }_{n=0}^{\infty }{x}_n(z-a{)}^n$ is a Banach space valued power series. Then either

• the series converges only for $z=a$ (formally $R=0$), or
• there exists a number $0<R<\infty$ such that the series converges absolutely whenever $|z-a|<R$ and diverges whenever $|z-a|>R$, or
• the series converges absolutely for all $z\in \mathbb{C}$ (formally $R=\infty$).

The number $R\in [0,\infty ]$ is called the radius of convergence of the power series.

Proof   TODO $\square$

Cauchy–Hadamard Formula

Let ${\sum }_{n=0}^{\infty }{x}_n(z-a{)}^n$ be a Banach space valued power series with radius of convergence $R$. Then

$$\frac{1}{R}=\underset{n\to \infty }{\mathrm{limsup}}\Vert x_n{\Vert }^{1/n}.$$

Proof   TODO $\square$