So my objective is to find the power series expansion of
$$\sin(2x)\cos(x)$$
This is what I know so far, I just need a little help find the series:
First of all, we know that ${\sin(2x)=\sum\limits_{n=0}^{\infty}\dfrac{(-1)^n}{(2n+1)!}x^{2n+1}}$ and that ${\cos(x)=\sum\limits_{n=0}^{\infty}\dfrac{(-1)^n}{(2n)!}x^{2n}}$.
My question is when I'm trying to find the series expansion of two functions, both with known series, that multiply each other, do we simply add or multiply the series expansion of each function.
Best Answer
Use the trigonometric sum-product relations to get something that just involves a difference of two sines:
$\sin(2x)\cos(x)=(1/2)(\sin(3x)+\sin(x))$