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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

  1. If \(L < 1\), then the series absolutely converges.
  2. If \(L > 1\), then the series diverges.
  3. If \(L = 1\), then the series is either divergent or convergen
Hence, the radius of convergence will be \(R = 1/L\).

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