Question

How do you find the taylor series for `f(x)=xcos(x^2)`?

Answer 1

The quickest way is to first recall the Taylor series for cosine about `x=0`:

`cos(x)=1-x^2/(2!)+x^4/(4!)-x^6/(6!)+\cdots`

Now just replace all these `x`'s by `x^2`'s and then multiply everything by `x`:

`xcos(x^2)=x-x^5/(5!)+x^9/(9!)-x^13/(6!)+\cdots`.

This can be shown to converge for all `x`.

You could also use the formula

`f(0)+f'(0)x+(f''(0))/(2!)x^2+(f'''(0))/(3!)x^3+\cdots`, but that would be a real big pain for this problem.