Definition:Power Series

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Definition

Real Domain

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

Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$.


The series $\displaystyle \sum_{n \mathop = 0}^\infty a_n \left({x - \xi}\right)^n$, 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 $\left \langle {a_n} \right \rangle$ be a sequence in $\C$.


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


Coefficient

The terms:

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

are the coefficients of $\displaystyle \sum_{n \mathop = 0}^\infty a_n \left({x - \xi}\right)^n$.


Center

The point $\xi$ is the center of $\displaystyle \sum_{n \mathop = 0}^\infty a_n \left({x - \xi}\right)^n$.


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