arcsine 'taylor expansion' general series

[Guest deenhead]

A question was posed in our calc class about how the general formula for the taylor series expansion of arcsin was created. Here is the general formula:

 

 

I tried calculating each derivative. But after the fifth one, it got too long and I still couldn't see a pattern. Here is a readout from the ti-89

 

f (0) = 0

f` (0) = 1

f`` (0) = 0

f```(0) = 1

f^4 (0) = 0

f^5 (0) = 9

f^6 (0) = 0

f^7 (0) = 225 = 15^2

f^8 (0) = 0

f^9 (0) = 105^2 ?

 

So I was wondering if any of you could help me in trying to find how the general formula was created.

 

thanks

[MandrakeRoot]

Let me help you a bit :

 

The general form of the n-th derivative of the arcsine function has the following form

[math]\frac{\partial f^{(n)}}{\partial x^n}(x) = (1 - x^2)^{-\frac{2n-1}{2}}p_n(x)[/math],

where p_n is a polynomial that satisfies the following

[math]p_{n+1}(x) = (2n-1)xp_n(x) + (1- x^2)p_n'(x)[/math],

Here n >= 1 and p_1 is all constant 1 function.

 

Mandrake