Calculating Taylor's Series for sin(x^2) - First Three Terms

[beanryu]

Find the first three terms of the Taylor's series for sin(x^2)

The constant term is 0 as usual. The first non-zero term is (include power of x as well as its coefficient):

according to the book

f(x)=f(0)+f'(0)x+f''(0)x^2/2!+f'''(0)x^3/3!...

so I think the first non zero term is

f'(0) = cos(x^2)2x

but substituting x with 0, the term will be zero

and there will always be an x in the expression if I keep differentiating.

So how do I do it?

THANKS!

 

[HallsofIvy]

f(x)= sin(x^2) so f'(x)= 2x cos(x^2) as you say. However, when you find the second derivative you use the product rule: f"(x)= 2 cos(x^2)- 4x^2 sin(x^2). No, there is not always an x in the expression. f"(0)= 2.

In fact, if you know the Taylor's series for sin(x) you can get the Taylors series for sin(x^2) just by substuting x^2 for x in it.

 

[beanryu]

Thanx alots!