# 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 $c_k$.
• @justin, Taylor polynomial $=$ partial sum Taylor series. – Martín-Blas Pérez Pinilla Oct 31 '14 at 14:43