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: $$r_n(x)=\frac{1}{n!}\int_{x_0}^x f^{(n+1)}(t)(x-t)^n dt.$$

Taylor's formula, with all forms of the remainder term given above, can be extended to the case of a function of several variables. Taylor's formula is also valid for mappings of subsets of a normed space into similar spaces, and in this case the remainder term can be written in Peano's form or in integral form.

Taylor's formula allows one to reduce the study of a number of properties of a function differentiable a specified number of times to the substantially simpler problems of studying these properties for the corresponding Taylor polynomial. This is the basis of various and numerous applications of the Taylor polynomial, for instance to the computation of limits of functions, to the investigation of their extreme points, their points of inflection, intervals of convexity and concavity, as well as to the convergence of series and integrals and to estimates of the speed of their convergence or divergence.

References

[1]	V.A. Il'in, V.A. Sadovnichii, B.Kh. Sendov, "Mathematical analysis" , Moscow (1979) (In Russian)
[2]	S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)

[a1]	T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1957)
[a2]	W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24
[a3]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78
[a4]	K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)