Power Series

A power series (in one variable) is an infinite series of the form:

 $\displaystyle{f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n = a_0 + a_1 (x-c)^1 + a_2 (x-c)^2 + \cdots}$

where $a_n$ represents the coefficient of the $n$th term, $c$ is a constant, and $x$ varies around $c$ (for this reason one sometimes speaks of the series as being centered at $c$). This series usually arises as the Taylor series of some known function. Any polynomial can be easily expressed as a power series around any center $c$, albeit one with most coefficients equal to zero. For instance, the polynomial

 $f(x) = x^2 + 2x + 3$

can be written as a power series around the center $c=1$ as:

 $f(x) = 6 + 4 (x-1) + 1(x-1)^2 + 0(x-1)^3 + 0(x-1)^4 + \cdots \,$

or, indeed, around any other center $c$.

Exponential Function as a Power Series

The exponential function (in blue), and the sum of the first $n+1$ terms of its Maclaurin power series (in red).

In many situations $c$ is equal to zero—for instance, when considering a Maclaurin series. In such cases, the power series takes the simpler form 

$\displaystyle{f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots}$

These power series arise primarily in real and complex analysis, but also occur in combinatorics (under the name of generating functions) and in electrical engineering (under the name of the $Z$-transform). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument $x$ fixed at $\frac{1}{10}$. In number theory, the concept of $p$-adic numbers is also closely related to that of a power series.

Radius of Convergence

A power series will converge for some values of the variable $x$ and may diverge for others. All power series $f(x)$ in powers of $(x-c)$ will converge at $x=c$. If $c$ is not the only convergent point, then there is always a number $r$ with 0 < r ≤ ∞ such that the series converges whenever $\left| x-c \right| <r$ and diverges whenever $\left| x-c \right| >r$. The number $r$ is called the radius of convergence of the power series. According to the Cauchy-Hadamard theorem, the radius $r$ can be computed from

$\displaystyle{r^{-1}=\lim_{n\to\infty}\left|{a_{n+1}\over a_n}\right|}$ 

if this limit exists.