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单词 PowerSeries
释义

power series


A power seriesMathworldPlanetmath is a series of the form

k=0ak(x-x0)k,

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:

  • Taylor seriesMathworldPlanetmath, for example:

    ex=k=0xkk!.
  • The geometric seriesMathworldPlanetmath:

    11-x=k=0xk,

    with |x|<1.

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:

    k=0ak(x-xo)kl=0bj(x-x0)j=a0b0+(a0b1+a1b0)(x-x0)+(a0b2+a1b1+a2b0)(x-x0)2.
  • (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.

Titlepower series
Canonical namePowerSeries
Date of creation2013-03-22 12:32:55
Last modified on2013-03-22 12:32:55
Ownerazdbacks4234 (14155)
Last modified byazdbacks4234 (14155)
Numerical id23
Authorazdbacks4234 (14155)
Entry typeDefinition
Classificationmsc 40A30
Classificationmsc 30B10
Related topicTaylorSeries
Related topicFormalPowerSeries
Related topicTermwiseDifferentiation
Related topicAbelsLimitTheorem
Definesconstant term
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更新时间:2025/5/4 12:18:53