Maclaurin series of $f(x)=x^3\sin 2x$

Question

I need help finding that maclaurin series for following function. $$f(x)= x^3 \sin2x$$ How do you get to the maclaurin series?

Answer 1

Expand the Maclaurin series for $\sin(2x)$ first.

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

Then the Maclaurin series for $f(x)$ is given by

$$f(x) = x^3\sin(2x) = \sum_{n=0}^\infty \frac{(-1)^n2^{2n+1}}{(2n+1)!} x^{2n+4}$$