Taylor series of arcus sine

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

 $\frac{d\,\arcsin{x}}{dx}=\frac{1}{\sqrt{1\!-\!x^{2}}}=(1\!-\!x^{2})^{-\frac{1}% {2}}.$

Using the generalized binomial coefficients ${-\frac{1}{2}\choose r}$ we thus can form the Taylor series for it as Newton’s binomial series (http://planetmath.org/BinomialFormula):

 $(1\!-\!x^{2})^{-\frac{1}{2}}=\sum_{r=0}^{\infty}{-\frac{1}{2}\choose r}(-x^{2}% )^{r}=1\!+\!{-\frac{1}{2}\choose 1}(-x^{2})\!+\!{-\frac{1}{2}\choose 2}(-x^{2}% )^{2}\!+\!{-\frac{1}{2}\choose 3}(-x^{2})^{3}\!+\!\cdots=$
 $=1\!-\!\frac{-\frac{1}{2}}{1!}x^{2}\!+\!\frac{-\frac{1}{2}(-\frac{1}{2}\!-\!1)% }{2!}x^{4}\!-\!\frac{-\frac{1}{2}(-\frac{1}{2}\!-\!1)(-\frac{1}{2}\!-\!2)}{3!}% x^{6}\!+-\cdots=$
 $=1\!+\!\frac{1}{2}x^{2}\!+\!\frac{1\cdot 3}{2\cdot 4}x^{4}\!+\!\frac{1\cdot 3% \cdot 5}{2\cdot 4\cdot 6}x^{6}\!+\!\cdots\quad\quad\quad\quad\mathrm{for}\,\,-% 1

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

 $\arcsin{x}=\int_{0}^{x}\frac{dx}{\sqrt{1\!-\!x^{2}}}=x\!+\!\frac{1}{2}\!\cdot% \!\frac{x^{3}}{3}\!+\!\frac{1\!\cdot\!3}{2\!\cdot\!4}\!\cdot\!\frac{x^{5}}{5}% \!+\!\frac{1\!\cdot\!3\cdot\!5}{2\!\cdot\!4\cdot\!6}\!\cdot\!\frac{x^{7}}{7}\!% +\!\cdots,$

the validity of which is true for  $|x|<1$.  It can be proved, in addition  , that it is true also when  $x=\pm 1$.

 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