Difference between revisions of "Taylor formula"

A representation of a function as a sum of its Taylor polynomial of degree $n$ ($n=0,1,2,\dotsc$) and a remainder term. If a real-valued function $f$ of one variable is $n$ times differentiable at a point $x_0$, its Taylor formula has the form $$f(x)=P_n(x)+r_n(x),$$ where $$P_n(x)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k$$ is its Taylor polynomial, while the remainder term $r_n(x)$ can be written in Peano's form: $$r_n(x)=o\left((x-x_0)^n\right),\quad x\to x_0.$$ If the function $f$ is $n+1$ times differentiable in some neighbourhood $(x_0-\delta,x_0+\delta)$, $\delta>0$, of a point $x_0$, then in this neighbourhood the remainder term can be written in the Schlömilch–Roch form $$r_n(x)=\frac{f^{(n+1)}(x_0+\theta(x-x_0)))}{n!p}(1-\theta)^{n-p+1}(x-x_0)^{n+1},$$ where $p=1,\dotsc,n+1$; as special cases there are the Lagrange form $$r_n(x)=\frac{f^{(n+1)}(x_0+\theta(x-x_0))}{(n+1)!}(x-x_0)^{n+1}$$ and the Cauchy form $$r_n(x)=\frac{f^{(n+1)}(x_0+\theta(x-x_0))}{n!}(1-\theta)^n(x-x_0)^{n+1},$$ $$0<\theta<1,\quad x\in(x_0-\delta,x_0+\delta);$$ the number $\theta$ depends on $x$, $p$ and $n$.
If the derivative of order $n+1$ of the function $f$ is integrable on the interval with end points $x$ and $x_0$, then the remainder term can be written in integral form: $$r_n(x)=\frac{1}{n!}\int_{x_0}^x f^{(n+1)}(t)(x-t)^n dt.$$