Find the Taylor series and prove it converges using the definition

Question

I'm studying for the FE Exam. A simple walk-through would be appreciated to help my understanding of how to solve similar problems.

Find the Taylor series about $x=2$ for the function

$f(x) = x^5 - 3x^4 + 2x^3 - x^2 - x + 1 $

And prove that the Taylor series converges to $f$ using the definition.

Answer 1

As $f^{(k)}=0$ for $k>5$, the Taylor series is a finite sum and $=f$.

Answer 2

Hint

As answered by Martín-Blas Pérez Pinilla, you just need to focus on the first six terms. You can apply the standard method computing the values of the function and the first fifth derivatives or replace in your expression $x$ by $(t+2)$ and expand. Doing so, you will arrive to $$t^5+7 t^4+18 t^3+19 t^2+3 t-5$$ that is to say that $$f(x) = -5+3 (x-2)+19 (x-2)^2+18 (x-2)^3+7 (x-2)^4+(x-2)^5$$