1
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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 $c_k$.

$\endgroup$
  • $\begingroup$ is there relation between Taylor polynomial and Taylor series. $\endgroup$ – justin Oct 31 '14 at 12:21
  • $\begingroup$ @justin, Taylor polynomial $=$ partial sum Taylor series. $\endgroup$ – Martín-Blas Pérez Pinilla Oct 31 '14 at 14:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.