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

May 22, 2015

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.