Power series

This article deals with power series, which are used to describe real or complex functions. For formal power series see there.

In calculus, a power series P(x) is an infinite series of the form

$P(x)=\sum _{{n=0}}^{\infty }a_{n}(x-x_{0})^{n}$

with

of any sequence $(a_{n})_{{n\in {\mathbb N}_{0}}}$ real or complex numbers.
the development point $x_{0}$ of the power series.

Power series play an important role in function theory and often allow a meaningful continuation of real functions into the complex number plane. In particular, the question arises for which real or complex numbers a power series converges. This question leads to the notion of the radius of convergence.

Radius of Convergence

→ Main article: Radius of convergence

The radius of convergence of a power series around the development point $x_{0}$ is defined to be the largest number for which the power series $|x-x_{0}|<r$ converges for all with . The open sphere $U_{r}(x_{0})$ with radius around $x_{0}$ is called a circle of convergence. Thus, the radius of convergence is the radius of the circle of convergence. If the series converges for all , the radius of convergence is said to be infinity. If it converges only for $x_{0}$ , then the radius of convergence is 0. The series is then sometimes called nowhere convergent.

For power series, the radius of convergence can be calculated using Cauchy-Hadamard's formula. It holds:

$r={\frac {1}{\limsup \limits _{n\rightarrow \infty }\ {\sqrt[{n}]{|a_{n}|}}}}$

In this context, one defines ${\tfrac {1}{0}}:=+\infty$ and ${\tfrac {1}{\infty }}:=0$

In many cases, the radius of convergence for power series with non-vanishing coefficients can also be calculated more simply. Namely it holds

$r=\lim _{n\to \infty }\left|{\frac {a_{n}}{a_{n+1}}}\right|,$

if this limit value exists.

Examples

Any polynomial function can be thought of as a power series where almost all coefficients $a_{n}$ are equal to 0. Important other examples are Taylor series and Maclaurin series. Functions that can be represented by a power series are also called analytic functions. Here is an example of the power series representation of some well-known functions:

Exponential function: $e^{x}=\exp(x)=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}={\frac {x^{0}}{0!}}+{\frac {x^{1}}{1!}}+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\dotsb$ for all $x\in \mathbb {R}$ , i.e., the radius of convergence is infinite.

Sine: $\sin(x)=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{2n+1}}{(2n+1)!}}={\frac {x}{1!}}-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}\mp \dotsb$

Cosine: $\cos(x)=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{2n}}{(2n)!}}={\frac {x^{0}}{0!}}-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}\mp \dotsb$

The radius of convergence is infinite for both the sine and the cosine. The power series representation results directly from the exponential function using Euler's formula.

Logarithm function: $\ln(1+x)=\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {x^{k}}{k}}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\dotsb$

for $-1 < x \leq 1$ , i.e.: The radius of convergence is 1. For x=1 the series is convergent, for x=-1 divergent.

Root function: ${\sqrt {1+x}}=1+{\frac {1}{2}}x-{\frac {1}{2\cdot 4}}x^{2}+{\frac {1\cdot 3}{2\cdot 4\cdot 6}}x^{3}\mp \dotsb$ for $-1\leq x\leq 1$ , d. h., the radius of convergence is 1 and the series converges both for and for .

Power series

Radius of Convergence

Examples

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