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:


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+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


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

Title Taylor series of arcus sine
Canonical name TaylorSeriesOfArcusSine
Date of creation 2013-03-22 14:51:18
Last modified on 2013-03-22 14:51:18
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Example
Classification msc 26A36
Classification msc 26A09
Classification msc 11B65
Classification msc 05A10
Related topic ExamplesOnHowToFindTaylorSeriesFromOtherKnownSeries
Related topic TaylorSeriesOfArcusTangent
Related topic CyclometricFunctions
Related topic LogarithmSeries