power series

Power sesries are series that contain a free parameter, or in other words, describe a function whenever they are convergent.

Power series

Let $(a_k{)}_{k=0}^{\infty }\subset C$ be a sequence and let $c\in \mathbb{C}$.
Then, for $z\in \mathbb{C}$, we define the (formal) power series as

$$f(z):=\sum _{k=0}^{\infty }{a}_k(z-c{)}^k$$

We call $a_k$ the coefficients and $c$ the center of the power series.
We call

$$R:=R(f):=\frac{1}{\lim {\sup }_{k\to \infty }\sqrt[k]{|a_k|}}$$

the radius of convergence of $f$.
We set $R=\infty$ if $\lim {\sup }_{k\to \infty }\sqrt[k]{|a_k|}=0$ and $R=0$ if $\lim {\sup }_{k\to \infty }\sqrt[k]{|a_k|}=\infty$.
We call $D_f=\{y\in \mathbb{C}:|<-c|<R(f)\}$ the disc of convergence of $f$.

Formal means we do not know a priori if the power series converges for a given $z$.

For real power series, $D_f=\{y\in \mathbb{R}:|<-c|<R(f)\}=(c-R,c+R)$ is called the interval of convergence.

Theorem 3.96, 3.98, Example 3.101, 3.104, intuition, …

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