So I was given the following prompt:
"For the following, use the definition to find the Taylor (or Maclaurin) series centered at c for the function. When writing your answers, be sure to list the first 4 non-zero terms and the general term."
$$f(x)=\sin(2x), \ \ \ c=0.$$
I guess I'm confused on how I'd find the general term from this, I worked out the Maclaurin series to look something like the following: $P_n(x)=2x-\frac{4}{3}x^3+\frac{4}{15}x^5-\frac{8}{315}x^7$. I understand that I needed to look for the pattern here to find the general term, but I don't know where I'd start for that and I also didn't know if there was an easier way to do this through a formula. Any help would be appreciated!
Best Answer
Your Maclauren series is correct. If you look at how you got the terms before you canceled the $2$s to reduce to lowest terms you can see the general term is $\pm \frac {2^n}{n!}$.
That doesn't get you a Taylor series centered at $c$. Clearly the constant term is $\sin 2c$ and you will have terms of all orders because the sine function is not odd around $c$ in general. You can either use the Taylor series formula or write $$\sin(2(c+x))=\sin(2c)\cos (2x) + \cos(2c) \sin (2x)$$
and use the Maclauren series for both sine and cosine modified as above for the factor $2$