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 derivative.

For    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 $\left(\genfrac{}{}{0pt}{}{-\frac{1}{2}}{r}\right)$ we thus can form the Taylor series for it as Newton’s binomial series (http://planetmath.org/BinomialFormula):

$${(1-x^2)}^{-\frac{1}{2}}=\sum _{r=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0pt}{}{-\frac{1}{2}}{r}\right){(-x^2)}^r=1+\left(\genfrac{}{}{0pt}{}{-\frac{1}{2}}{1}\right)(-x^2)+\left(\genfrac{}{}{0pt}{}{-\frac{1}{2}}{2}\right){(-x^2)}^2+\left(\genfrac{}{}{0pt}{}{-\frac{1}{2}}{3}\right){(-x^2)}^3+\mathrm{\cdots }=$$

$$=1-\frac{-\frac{1}{2}}{1!}{x}^2+\frac{-\frac{1}{2}(-\frac{1}{2}-1)}{2!}{x}^4-\frac{-\frac{1}{2}(-\frac{1}{2}-1)(-\frac{1}{2}-2)}{3!}{x}^6+-\mathrm{\cdots }=$$

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

$$\arcsin x={\int }_0^x\frac{dx}{\sqrt{1-x^2}}=x+\frac{1}{2}\cdot \frac{x^3}{3}+\frac{1\cdot 3}{2\cdot 4}\cdot \frac{x^5}{5}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\cdot \frac{x^7}{7}+\mathrm{\cdots },$$

the validity of which is true for  .  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