Dervation

Taylor Series

Keywords: taylor series, remainder, Lagrange

The Taylor series of a function f (x), that is infinitely differentiable at number a, is the power series:

$T\left(x\right)=\sum _{n=0}^{\infty }\frac{{f}^{\left(n\right)}\left(a\right)}{n!}{\left(x-a\right)}^{n}$

or in other form with (n-1) term and the remainder:

$T\left(x\right)=\sum _{k=0}^{n-1}\frac{{f}^{\left(k\right)}\left(a\right)}{k!}{\left(x-a\right)}^{k}+{R}_{n}$

Lagrange's form

${R}_{n}=\frac{{f}^{\left(n\right)}\left(\xi \right){\left(x-a\right)}^{n}}{n!}$

Cauchy's form

${R}_{n}=\frac{{f}^{\left(n\right)}\left(\xi \right){\left(x-\xi \right)}^{n-1}\left(x-a\right)}{\left(n-1\right)!}$