Taylor Expansion

**jzellt** · December 8th 2009, 07:56 PM

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.

**Prove It** · December 8th 2009, 08:16 PM

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...

**HallsofIvy** · December 9th 2009, 05:44 AM

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

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