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单词 TaylorSeriesOfArcusTangent
释义

Taylor series of arcus tangent


The derivativePlanetmathPlanetmath of the arcus tangentPlanetmathPlanetmathPlanetmath (http://planetmath.org/CyclometricFunctions) function, 11+x2, can be expanded as a geometric seriesMathworldPlanetmath

11+x2=1-x2+x4-x6+-,

the radius of convergenceMathworldPlanetmath of which is 1.  The series is valid only on the open intervalPlanetmathPlanetmath-1<x<1,  because the series apparently divergesPlanetmathPlanetmath for  x=±1.  The power seriesMathworldPlanetmath may be integrated termwise on its interval of convergence, giving

0xdt1+t2=/0xarctant=arctanx=x-x33+x55-+  (|x|<1).

We can show that this Taylor series of arcus tangent is valid also for the end points  x=±1  of the interval.

We start from the identical equation

11+t2=1-t2+t4-++(-1)n-1t2n-2+(-1)nt2n1+t2,

which can be verified by performing the division 1:(1+t2).  Integrating both sides from 0 to an arbitrary x, we obtain

arctanx=x-x33+x55-++(-1)n-1x2n-12n-1S2n-1+(-1)n0xt2n1+t2𝑑tR2n-1.

We estimate R2n-1:

|R2n-1|=0|x|t2n1+t2𝑑t0|x|t2n𝑑t=|x|2n-12n-1 0asn

for  x=±1.  Accordingly, when  x=±1,  we see that

S2n-1=arctanx-R2n-1arctanx

as  n.  This that

arctan(±1)=±π4=±(1-13+15-+).
TitleTaylor series of arcus tangent
Canonical nameTaylorSeriesOfArcusTangent
Date of creation2013-03-22 16:33:37
Last modified on2013-03-22 16:33:37
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id7
Authorpahio (2872)
Entry typeDerivation
Classificationmsc 30B10
Classificationmsc 26A24
Classificationmsc 41A58
Related topicGregorySeries
Related topicTaylorSeriesOfArcusSine
Related topicSubstitutionNotation
Related topicExamplesOnHowToFindTaylorSeriesFromOtherKnownSeries
Related topicCyclometricFunctions
Related topicLogarithmSeries
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