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.


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.


1 Answer 1


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 $c_k$.

  • $\begingroup$ is there relation between Taylor polynomial and Taylor series. $\endgroup$
    – justin
    Oct 31, 2014 at 12:21
  • $\begingroup$ @justin, Taylor polynomial $=$ partial sum Taylor series. $\endgroup$ Oct 31, 2014 at 14:43

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