Taylor series of arcus tangent

The derivative of the arcus tangent (http://planetmath.org/CyclometricFunctions) function, $\frac{1}{1+x^2}$, can be expanded as a geometric series

$$\frac{1}{1+x^2}=1-x^2+x^4-x^6+-\mathrm{\dots },$$

the radius of convergence of which is $1$.  The series is valid only on the open interval  ,  because the series apparently diverges for  $x=\pm 1$.  The power series may be integrated termwise on its interval of convergence, giving

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

We start from the identical equation

$$\frac{1}{1+t^2}=1-t^2+t^4-+\mathrm{\dots }+{(-1)}^{n-1}{t}^{2n-2}+{(-1)}^n\frac{t^{2n}}{1+t^2},$$

which can be verified by performing the division $1:(1+t^2)$.  Integrating both sides from $0$ to an arbitrary $x$, we obtain

$$\arctan x=\underset{S_{2n-1}}{\underbrace{x-\frac{x^3}{3}+\frac{x^5}{5}-+\mathrm{\dots }+{(-1)}^{n-1}\frac{x^{2n-1}}{2n-1}}}+\underset{R_{2n-1}}{\underbrace{{(-1)}^n{\int }_0^x\frac{t^{2n}}{1+t^2}𝑑t}}.$$

We estimate $R_{2n-1}$:

$$|R_{2n-1}|={\int }_0^{|x|}\frac{t^{2n}}{1+t^2}𝑑t\leqq {\int }_0^{|x|}{t}^{2n}𝑑t=\frac{{|x|}^{2n-1}}{2n-1}\to \mathrm{\hspace{0.17em}0}\mathit{\hspace{1em}}\mathrm{as}n\to \mathrm{\infty }$$

for  $x=\pm 1$.  Accordingly, when  $x=\pm 1$,  we see that

$$S_{2n-1}=\arctan x-R_{2n-1}\to \arctan x$$

as  $n\to \mathrm{\infty }$.  This that

$$\arctan (\pm 1)=\pm \frac{\pi }{4}=\pm (1-\frac{1}{3}+\frac{1}{5}-+\mathrm{\dots }).$$

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