Power Series

Definition

Given a sequence $\{c_n\}$ of Complex Numbers, the Series,

$$\sum _{n=0}^{\infty }{c}_n{z}^n$$

is called a power series. The numbers $c_n$ are the coefficients of the series, and $z\in \mathbb{C}$. Depending on the choice of $z$ this power series will converge or diverge.

Radius of Convergence

Given the power series $\sum c_n{z}^n$ put

$$\alpha =\underset{n\to \infty }{\mathrm{limsup}}\sqrt[n]{|c_n|}R=\frac{1}{\alpha }$$

Where if $\alpha =0$ then $R=+\infty$ and $\alpha =+\infty$ then $R=0$. Then $\sum c_n{z}^n$ converges if $|z|<R$ and diverges if $|z|>R$.

Proof: Let $a_n=c_n{z}^n$, and apply root test:

$$\underset{n\to \infty }{\mathrm{limsup}}\sqrt[n]{|a_n|}=|z|\mathrm{limsup}\sqrt[n]{|c_n|}=\frac{|z|}{R}$$

and $R$ is the radius of convergence.

Special Radii

1. $\sum n^n{z}^n$ has $R=0$
2. $\sum z^n/n!$ has $R=+\infty$. Here we can use the ratio test. By Ratio-Root Relationship we can obtain the same radius.
3. $\sum z_n$ has $R=1$. If $|z|=1$ then it diverges.
4. $\sum z^n/n$ has $R=1$. it diverges if $z=1$. It converges for all other $z$ with $|z|=1$.