# Taylor series of arcus tangent

 $\frac{1}{1\!+\!x^{2}}=1\!-\!x^{2}\!+\!x^{4}\!-\!x^{6}\!+-\ldots,$

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

 $\int_{0}^{x}\!\frac{dt}{1\!+\!t^{2}}=\operatornamewithlimits{\Big{/}}_{\!\!\!0% }^{\,\quad x}\!\arctan{t}=\arctan{x}=x\!-\!\frac{x^{3}}{3}\!+\!\frac{x^{5}}{5}% \!-+\ldots\qquad(|x|<1).$

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

 $\frac{1}{1\!+\!t^{2}}=1\!-\!t^{2}\!+\!t^{4}\!-+\ldots+\!(-1)^{n-1}t^{2n-2}+(-1% )^{n}\frac{t^{2n}}{1+t^{2}},$

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

 $\arctan{x}=\underbrace{x\!-\!\frac{x^{3}}{3}\!+\!\frac{x^{5}}{5}\!-+\ldots+\!(% -1)^{n-1}\frac{x^{2n-1}}{2n\!-\!1}}_{S_{2n-1}}+\underbrace{(-1)^{n}\int_{0}^{x% }\frac{t^{2n}}{1+t^{2}}dt}_{R_{2n-1}}.$

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

 $|R_{2n-1}|=\int_{0}^{|x|}\frac{t^{2n}}{1+t^{2}}dt\leqq\int_{0}^{|x|}t^{2n}\,dt% =\frac{|x|^{2n-1}}{2n\!-\!1}\,\to\,0\quad\mathrm{as}\,\,n\to\infty$

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

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

as  $n\to\infty$.  This that

 $\arctan{(\pm 1)}=\pm\frac{\pi}{4}=\pm\left(1\!-\!\frac{1}{3}\!+\!\frac{1}{5}\!% -+\ldots\right).$
 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