MHB Jason Z's question at Yahoo Answers (Maclaurin series cuestion)

[Fernando Revilla]

Here is the question:

Find the coefficient of x^3 in the macluarin series for sinxcosx

show work please and thank you!

Here is a link to the question:

Maclaurin Series Question? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello Jason Z,

We have $\sin x\cos x=\dfrac{1}{2}\sin 2x$. On the other hand for all $t\in\mathbb{R}$, $\sin t= t-\dfrac{t^3}{3!}+\ldots$ so $$\sin x\cos x=\frac{1}{2}\left(2x-\frac{(2x)^3}{3!}+\ldots\right)\Rightarrow \mbox{coef. }(x^3)=\frac{1}{2}\cdot \frac{8}{6}=\frac{2}{3}$$