Maclaurin series question

Question

What is the Maclaurin series of $$z^3\sin(z^2)$$

I'm able to differentiate it but can't figure out how to write it in general series form, can somebody help me please.

Answer 1

Recall that

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

Thus,

$$\sin(z^2)=\sum_{n=0}^\infty\frac{(-1)^nz^{4n+2}}{(2n+1)!}$$

and finally

$$z^3\sin(z^2)=\sum_{n=0}^\infty\frac{(-1)^nz^{4n+5}}{(2n+1)!}$$