Associated Topics || Dr. Math Home || Search Dr. Math

Taylor Expansion

```
Date: 11/21/2001 at 05:16:59
Subject: Taylor Expansion

Can you give me the proof of this statement?

The basis of the calculation is a Taylor series:

arcsin(x) = x + 1/2 (x^3/3) + (1/2)(3/4)(x^5/5) +
(1/2)(3/4)(5/6)(x^7/7) + ...

Best regards,
Henning
```

```
Date: 11/21/2001 at 12:10:17
From: Doctor Pete
Subject: Re: Taylor Expansion

Hi,

This series can be found by using the generalized Binomial Theorem,
which says

(1+x)^n = a[0] + a[1]x + a[2]x^2 + ... + a[k]x^k + ...

where

a[k] = Binomial[n,k] = n!/(k!(n-k)!).

However, n is not an integer. Since the derivative of the arcsine is

(1-x^2)^(-1/2),

we see that n = -1/2, and we substitute -x^2 for x in the formula.
Then integrating term by term gives the final result. You may wonder
how to define the binomial coefficient when n is not an integer.
Since

n!/(k!(n-k)!) = n(n-1)(n-2)...(n-k+1)/k!,

it is natural to use the right-hand side when n is not an integer.
For n = -1/2, we have

Binomial[1/2,k] = (-1/2)(-3/2)(-5/2)...(-(2k-1)/2)/k!
= (-1)^k (1*3*5*...*(2k-1))/(k!2^k).

We can express this as factorials of integers by noting

(2k)! = (1*3*5*...*(2k-1))(2*4*6*...*(2k))
= (1*3*5*...*(2k-1))(2^k)(1*2*3*...*k)
= (1*3*5*...*(2k-1))(2^k)k!,

therefore

a[k] = (-1)^k (2k)!/((2^k)k!)/(k!2^k)
= (-1)^k (2k)!/(k!2^k)^2.

It follows that the k(th) term of the expansion of (1-x^2)^(-1/2) is

a[k](-x^2)^k = (2k)!/(k!2^k)^2 x^(2k).

Integrating with respect to x, we obtain the k(th) term of the
expansion of arcsin(x),

(2k)!/(k!2^k)^2 x^(2k+1)/(2k+1),

where k = 0 to infinity.  It is not difficult to see that this agrees
with the orignal statement of the problem. By the uniqueness of the
series expansion, we can be assured that this is the series
representation of arcsin(x).

It is interesting to note that we did not rely on Taylor's formula,
which says that for F[x] = arcsin(x),

a[k] = F(k)[0]/k!,

where F(k)[x] is the k(th) derivative of F, evaluated at x.

- Doctor Pete, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory
High School Sequences, Series

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search