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


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 pointsx=±1  of the interval.

We start from the identical equation


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


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


as  n.  This that

Title Taylor series of arcus tangent
Canonical name TaylorSeriesOfArcusTangent
Date of creation 2013-03-22 16:33:37
Last modified on 2013-03-22 16:33:37
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Derivation
Classification msc 30B10
Classification msc 26A24
Classification msc 41A58
Related topic GregorySeries
Related topic TaylorSeriesOfArcusSine
Related topic SubstitutionNotation
Related topic ExamplesOnHowToFindTaylorSeriesFromOtherKnownSeries
Related topic CyclometricFunctions
Related topic LogarithmSeries