Newton's method of infinite series

hunkydory19 · #1 $Old$ November 6th, 2009, 01:10 PM

Given the infinite series representation for $sin^-1$ ,

$sin^-1x = x + \frac{x^3}{6} + \frac{3x^5}{40} + \frac{5x^7}{112} + ... = y$

apply Newton's Method for inverting infinite series to recover the first three non-zero terms of the series for $x = sin y.$

I supposed $x = a_0 + a_1y + a_2y^2 + a_3y^3 + a_4y^4 + a_5y^5$

substituted this into y, and then equated coefficients to find $a_1, a_2...a_5.$

The answer I got was $x = y - \frac{1}{6}y^3 + \frac{1}{120}y^5$

But I was wondering if anyone knows a way to check this answer, and also a quicker way of doing this (still using Newtons method) as I ended up expanding about 100 terms towards the end!

Thanks in advance!