# Definition:Power Series

## Definition

### Real Domain

Let $\xi \in \R$ be a real number.

Let $\sequence {a_n}$ be a sequence in $\R$.

The series $\displaystyle \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$, where $x \in \R$ is a variable, is called a **power series in $x$ about the point $\xi$**.

### Complex Domain

Let $\xi \in \C$ be a complex number.

Let $\sequence {a_n}$ be a sequence in $\C$.

The series $\displaystyle \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$, where $z \in \C$ is a variable, is called a **(complex) power series in $z$ about the point $\xi$**.

## Coefficient

The terms:

- $a_0, a_1, \ldots, a_n, \ldots$

are the **coefficients** of $\ds \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$.

## Center

The point $\xi$ is the **center** of $\ds \sum_{n \mathop = 0}^\infty a_n \size {x - \xi}^n$.

## Also see

- Results about
**power series**can be found here.

## Historical Note

Power series were studied by Karl Weierstrass, during the course of which he developed the concept of uniform convergence.

## Sources

- 1992: Larry C. Andrews:
*Special Functions of Mathematics for Engineers*(2nd ed.) ... (previous) ... (next): $\S 1.3.2$: Power series: $(1.39)$