power series

A power seriesMathworldPlanetmath is a series of the form


with ak,x0 or . The ak are called the coefficients and x0 the center of the power series. a0 is called the constant term.

Where it convergesPlanetmathPlanetmath the power series defines a function, which can thus be represented by a power series. This is what power series are usually used for. Every power series is convergent at least at x=x0 where it converges to a0. In addition it is absolutely and uniformly convergent in the region {x|x-x0|<r}, with

r=lim infk1|ak|k

It is divergent for every x with |x-x0|>r. For |x-x0|=r no general predictions can be made. If r=, the power series converges absolutely and uniformly for every real or complex x. The real number r is called the radius of convergenceMathworldPlanetmath of the power series.

Examples of power series are:

Power series have some important :

  • If a power series converges for a z0 then it also converges for all z with |z-x0|<|z0-x0|.

  • Also, if a power series diverges for some z0 then it diverges for all z with |z-x0|>|z0-x0|.

  • For |x-x0|<r Power series can be added by adding coefficients and multiplied in the obvious way:

  • (Uniqueness) If two power series are equal and their are the same, then their coefficients must be equal.

  • Power series can be termwise differentiated and integrated. These operationsMathworldPlanetmath keep the radius of convergence.

Title power series
Canonical name PowerSeries
Date of creation 2013-03-22 12:32:55
Last modified on 2013-03-22 12:32:55
Owner azdbacks4234 (14155)
Last modified by azdbacks4234 (14155)
Numerical id 23
Author azdbacks4234 (14155)
Entry type Definition
Classification msc 40A30
Classification msc 30B10
Related topic TaylorSeries
Related topic FormalPowerSeries
Related topic TermwiseDifferentiation
Related topic AbelsLimitTheorem
Defines constant term