Basic Manipulations

If we know a power series for a function $f(x)$ we can always find a power series for the function $f(u)$ where $u$ is some function of $x$. For example knowing

$$\sin(x)=\displaystyle \sum^{\infty}_{n=0}(-1)^n\displaystyle \frac{x^{2n+1}}{(2n+1)!}$$

we know that

$$\sin(x^2)=\displaystyle \sum^{\infty}_{n=0}(-1)^n \displaystyle \... ... \displaystyle \sum^{\infty}_{n=0}(-1)^n \displaystyle \frac{x^{4n+2}}{(2n+1)!}$$

Similarly, we can define a series for $x\cos(x)$. It is just

$$x\cos(x) = x\displaystyle \sum^{\infty}_{n=0}(-1)^n\displaystyle \frac{x^{2n}}{(2n)!}$$

$$= x\left(1-\displaystyle \frac{x^2}{2!}+\displaystyle \frac{x^4}{4!}-\displaystyle \frac{x^6}{6!}+\dots\right)$$

$$= x-\displaystyle \frac{x^3}{2!}+\displaystyle \frac{x^5}{4!}-\displaystyle \frac{x^7}{6!}+\dots$$

$$= \displaystyle \sum^{\infty}_{n=0}(-1)^n\displaystyle \frac{x^{2n+1}}{(2n)!}$$