Taylor formula with integral remainder

Taylor公式

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For an integer m>=0 and a real C^{m+1} function on an interval containing a and x, Taylor's formula writes f(x)=sum_{k=0}^m f^{(k)}(a)(x-a)^k/k! + (1/m!) integral_a^x (x-t)^m f^{(m+1)}(t) dt.

实函数在包含 a 和 x 的区间上具有连续的 m+1 阶导数时，可表示为 a 处直到 m 阶的 Taylor 多项式加上由 m+1 阶导数积分构成的余项。

Example

For $f(x)=e^x$ at $a=0$, $e^x=1+x+x^2/2+R_2(x)$, where $|R_2(x)|\le e^{|x|}|x|^3/6$.

Usage and distinctions

This is a finite expansion with a controlled remainder; smoothness alone does not guarantee equality with an infinite Taylor series.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Michael Taylor — Introduction to Analysis in Several Variables (Advanced Calculus)§2.1, equations (2.1.39)–(2.1.40), printed p.46 · Checked 2026-10-14

Translation sources

1. 北京大学数学科学学院 — 数学分析（实验班）课程介绍第一部分 一元微积分学，五、导数的应用：Taylor公式 · Checked 2026-10-14

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.