Definition of Power Series A power series is an infinite series of the form: $$\sum_{n=0}^{\infty} a_n (x - a)^n$$ where: \(a_n\) are complex numbers \(...
Definition of Power Series
A power series is an infinite series of the form:
where:
(a_n) are complex numbers
(x) is a real variable
The power series converges for all (x) in the domain of convergence, which is the set of all real numbers (x) for which the series converges. The power series is absolutely convergent within its domain of convergence and converges uniformly outside its domain of convergence.
Examples
The series (sum_{n=0}^{\infty} \frac{(-1)^n}{n!}) converges for all (x), since the absolute value of the terms decreases faster than (\frac{1}{n!}) as (n) increases.
The series (sum_{n=0}^{\infty} (2x)^n) converges for (x \ne 0), since the absolute value of the terms decreases faster than (\frac{1}{n}) as (n) increases.
The series (sum_{n=0}^{\infty} \frac{1}{n!}) diverges for all (x), since the absolute value of the terms grows infinitely as (n) increases.
Radius of Convergence
The radius of convergence of a power series is the radius of the circle centered at the origin in the complex plane. The radius of convergence of a power series is determined by the largest value of (r) such that the series converges absolutely when (|x| < r) and diverges when (|x| > r).
For power series, the radius of convergence is equal to the radius of convergence of the corresponding geometric series. Therefore, the radius of convergence of a power series is the same as the radius of convergence of the geometric series