My teacher gave us a study guide to work on, and one of the problems doesn't seem to come out right. The directions are to "find the Taylor series of $f(x)=x^5-3x^4+x^3+2x-1$ for $a=1$. I calculated the n-th derivatives and wound up with:
$f(1)=0; f'(1)=-2; f''(1)=-10; f'''(1)=-6; f^{(4)}(1)=48; f^{(5)}(1)=120 $
All derivatives afterward are 0.
In order to form the series, I tried plugging these in, but I got this:
$f(x)=(0) + (-2)(x-1) + \frac{(-10)(x-1)^2}{2!} + \frac{(-6)(x-1)^3}{3!} + \frac{48(x-1)^4}{4!} + \frac{120(x-1)^5}{5!}$
So far, I can't find a relationship between the coefficients, even simplifying the factorial. I'm also not sure how I'm supposed to alternate the sign, seeing as it doesn't change predictably. How do I solve this?