Is there any realtionship between linear approximation and taylor series? As said  at Where did the linear approximation/linearization formula come from? about linear approximation is there any thing that relates taylor series and linear approximation.
$f(x)$=$f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots$
As seen in the above equation does it shows that taylor approximated a function using derivatives of points,like if there are points $(x_0,f(x_0))$ and $(x_1,f(x_1))$ then as said in above referred webpage:$f(x_1)=f(x_0)+f'(x_0)(x_1-x_0)$(linear approximation).
But I couldn't get how the factorials came and how upto n derivatives should be taken to approximate a function.

The definition of Taylor polynomial: is the only polynomial of degree $n$ that coincides with the function and their derivatives up to the $n$-th:
$$P_n(a)=f(a)$$
$$P_n'(a)=f'(a)$$
$$P_n''(a)=f''(a)$$
$$\cdots$$
$$P_n^{(n)}(a)=f^{(n)}(a)$$
Write
$$P_n(x)=c_0+c_1(x-a)+\cdots c_n(x-a)^n,$$
and impose the $(n+1)$ conditions to find the $c_k$.
