# Power Series¶

A series of the form of

$$ f(z) = \sum_{n=0}^\infty a_n (z-a)^n \quad a_n \in \mathbb{C}$$ is called a power series. Here, \(a\) is called the center of the power series.

## 1. Radius of Convergence¶

Definition

For any power series there exist a real number \(R\geq 0\), such that, the given power series converges in open disc of radius \(R\), i.e., \(\vert z - a \vert < R\), while diverges for all \(z\) outside the closed disc of radius \(R\), i.e., \(\vert z - a \vert > R\). Such \(R\) is called the radius of convergence the given power series.

Warning

At the circle of convergence, i.e., at \(\vert z - a\vert = R\), the power series can converge for some \(z\) or diverge for some other points. For example consider the power series, $$ f(z) = \sum_{n=0}^\infty \dfrac{z^n}{n}, $$ the radius of convergence will be \(1\). It diverge for \(z=1\) but converge for \(z=-1\), both lie on the circle of convergence.

## 2. Method of finding Radius of Convergence¶

Two theorem from real analysis usualy used to find the radius of convergence as mentioned below.

Ratio Test

For the power series \(\sum_{n=0}^\infty a_n (z-a)^n\), consider the followin limit exist, $$ L = \lim_{n\to \infty}\left \vert \dfrac{a_n}{a_{n+1}}\right\vert.$$ Then

- If \(L < 1\), then the series absolutely converges.
- If \(L > 1\), then the series diverges.
- If \(L = 1\), then the series is either divergent or convergen

Root Test/Cauchy Hadamard Test

For the power series \(\sum_{n=0}^\infty a_n (z-a)^n\), the radius of converge is given by \(\(\dfrac{1}{R} = \limsup_{n\to \infty} \left\vert a_n \right \vert ^{\dfrac{1}{n}}\)\)

Last modified on: 2023-01-04 23:20:05