What is the Taylor Expansion of:
sin(sinx) at x=0 up to terms of order 3 ?
I know that the Taylor of sin(x) is: x - x^3/3 + ...
But then what is the Taylor of sin(x - x^3/3 + ...) ?
Thanks.
$\displaystyle \sin{X} = X - \frac{X^3}{3!} + \frac{X^5}{5!} - \frac{X^7}{7!} + \dots - \dots$.
In your case, $\displaystyle X = \sin{x}$.
So $\displaystyle \sin{(\sin{x})}$
$\displaystyle = \left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots - \dots\right)$
$\displaystyle - \frac{\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots - \dots\right)^3}{3!}$
$\displaystyle + \frac{\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots - \dots\right)^5}{5!}$
$\displaystyle - \frac{\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots - \dots\right)^7}{7!} + \dots - \dots$
Very long I know, and some simplifying needs doing as well...