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[parent] Taylor series of arcus sine (Example)

We give an example of obtaining the Taylor series expansion of an elementary function by integrating the Taylor series of its derivative.

For $ -1 < x < 1$ we have the derivative of the principal branch of the arcus sine function:

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

Using the generalized binomial coefficients $ {-\frac{1}{2} \choose r}$ we thus can form the Taylor series for it as Newton's binomial series:

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

Because $ \arcsin{0} = 0$ for the principal branch of the function, we get, by integrating the series termwise, the expansion

$\displaystyle \arcsin{x} = \int_0^x\frac{dx}{\sqrt{1\!-\!x^2}} = x\!+\!\frac{1}... ...rac{1\!\cdot\!3\cdot\!5}{2\!\cdot\!4\cdot\!6}\!\cdot\!\frac{x^7}{7}\!+\!\cdots,$
the validity of which is true for $ \vert x\vert < 1$. It can be proved, in addition, that it is true also when $ x = \pm 1$.



"Taylor series of arcus sine" is owned by pahio.
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See Also: examples on how to find Taylor series from other known series, Taylor series of arcus tangent


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Cross-references: addition, generalized binomial coefficients, function, derivative, elementary function, Taylor series

This is version 9 of Taylor series of arcus sine, born on 2004-11-24, modified 2006-09-24.
Object id is 6527, canonical name is TaylorSeriesOfArcusSine.
Accessed 5297 times total.

Classification:
AMS MSC: 05A10 (Combinatorics :: Enumerative combinatorics :: Factorials, binomial coefficients, combinatorial functions)
 11B65 (Number theory :: Sequences and sets :: Binomial coefficients; factorials; $q$-identities)
 26A09 (Real functions :: Functions of one variable :: Elementary functions)
 26A36 (Real functions :: Functions of one variable :: Antidifferentiation)
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