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单词 TaylorSeriesOfArcusSine
释义

Taylor series of arcus sine


We give an example of obtaining the Taylor series of an elementary function by integrating the Taylor series of its derivativePlanetmathPlanetmath.

For  -1<x<1  we have the derivative of the principal of the arcus sine (http://planetmath.org/CyclometricFunctions) function:

darcsinxdx=11-x2=(1-x2)-12.

Using the generalized binomial coefficients (-12r) we thus can form the Taylor series for it as Newton’s binomial series (http://planetmath.org/BinomialFormula):

(1-x2)-12=r=0(-12r)(-x2)r=1+(-121)(-x2)+(-122)(-x2)2+(-123)(-x2)3+=
=1--121!x2+-12(-12-1)2!x4--12(-12-1)(-12-2)3!x6+-=
=1+12x2+1324x4+135246x6+    for-1<x<1

Because  arcsin0=0  for the principal branch (http://planetmath.org/GeneralPower) of the function, we get, by integrating the series termwise (http://planetmath.org/SumFunctionOfSeries), the

arcsinx=0xdx1-x2=x+12x33+1324x55+135246x77+,

the validity of which is true for  |x|<1.  It can be proved, in additionPlanetmathPlanetmath, that it is true also when  x=±1.

TitleTaylor series of arcus sine
Canonical nameTaylorSeriesOfArcusSine
Date of creation2013-03-22 14:51:18
Last modified on2013-03-22 14:51:18
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id12
Authorpahio (2872)
Entry typeExample
Classificationmsc 26A36
Classificationmsc 26A09
Classificationmsc 11B65
Classificationmsc 05A10
Related topicExamplesOnHowToFindTaylorSeriesFromOtherKnownSeries
Related topicTaylorSeriesOfArcusTangent
Related topicCyclometricFunctions
Related topicLogarithmSeries
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更新时间:2025/5/4 18:53:15