Math Help - Taylor Expansion

1. Taylor Expansion

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.

2. Originally Posted by jzellt
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.
$\sin{X} = X - \frac{X^3}{3!} + \frac{X^5}{5!} - \frac{X^7}{7!} + \dots - \dots$.

In your case, $X = \sin{x}$.

So $\sin{(\sin{x})}$

$= \left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots - \dots\right)$

$- \frac{\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots - \dots\right)^3}{3!}$

$+ \frac{\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots - \dots\right)^5}{5!}$

$- \frac{\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots - \dots\right)^7}{7!} + \dots - \dots$

Very long I know, and some simplifying needs doing as well...

3. The fact that you only need "to order 3" simplifies a lot!