# 泰勒公式

（重定向自泰勒定理

## 泰勒公式

${\displaystyle {\textrm {e}}^{x}\approx 1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots +{\frac {x^{n}}{n!}}.}$

${\displaystyle R_{n}(x)={\textrm {e}}^{x}-\left(1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots +{\frac {x^{n}}{n!}}\right).}$

## 泰勒定理

${\displaystyle f(a+h)=f(a)+f^{\prime }(a)h+o(h)}$ ，其中${\displaystyle o(h)}$  是比h 高阶的无穷小

n 是一个正整数。如果定义在一个包含 a区间上的函数 fa 点处 n+1 次可导，那么对于这个区间上的任意 x，都有：

${\displaystyle f(x)=f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f^{(2)}(a)}{2!}}(x-a)^{2}+\cdots +{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}+R_{n}(x).}$ [2]

${\displaystyle R_{n}(x)}$  的表达形式有若干种，分别以不同的数学家命名。

${\displaystyle f(x)=f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f^{(2)}(a)}{2!}}(x-a)^{2}+\cdots +{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}+o[(x-a)^{n}]}$

${\displaystyle f(x)=f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f^{(2)}(a)}{2!}}(x-a)^{2}+\cdots +{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}+{\frac {f^{(n+1)}(\xi )}{(n+1)!}}(x-a)^{(n+1)}}$

${\displaystyle R_{n}(x)={\frac {f^{(n+1)}(\xi )}{(n+1)!}}(x-a)^{(n+1)}}$ ，其中${\displaystyle \xi \in (a,x)}$ [4]

${\displaystyle R_{n}(x)=\int _{a}^{x}{\frac {f^{(n+1)}(t)}{n!}}(x-t)^{n}\,dt,}$

## 余项估计

${\displaystyle f(x)=f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f^{(2)}(a)}{2!}}(x-a)^{2}+\cdots +{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}+R_{n}(x),}$

## 多元泰勒公式

${\displaystyle f(x)=\sum _{|\alpha |=0}^{n}{\frac {1}{\alpha !}}{\frac {\partial ^{\alpha }f(a)}{\partial x^{\alpha }}}(x-a)^{\alpha }+\sum _{|\alpha |=n+1}R_{\alpha }(x)(x-a)^{\alpha }}$

${\displaystyle x=(x_{1},x_{2},...,x_{n})}$  ,则记 :${\displaystyle x^{\alpha }=x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}...x_{n}^{\alpha _{n}}}$ ,${\displaystyle {\frac {\partial ^{\alpha }f(a)}{\partial x^{\alpha }}}={\frac {\partial ^{\alpha _{1}+\alpha _{2}+...+\alpha _{n}}f(a)}{\partial x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}...x_{n}^{\alpha _{n}}}}}$ .

${\displaystyle f(a+h)=\sum _{k=0}^{m}{\frac {\partial ^{\alpha }f(a)}{\alpha !}}{\partial x^{\alpha }}h^{\alpha }+R_{m}}$

${\displaystyle f(a+h)=f(a)+{\frac {\partial f}{\partial x_{1}}}(a)h_{1}+...+{\frac {\partial f}{\partial x_{n}}}(a)h_{n}+{\frac {1}{2}}\sum _{i,j=1}^{n}{\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}(a)+...}$ .

${\displaystyle f(a+h)=f(a)+Jf(a)h+{\frac {1}{2}}(h_{1},...h_{n})Hf(a){\begin{pmatrix}h_{1}\\...\\h_{n}\\\end{pmatrix}}+...}$

## 参考来源

1. ^ J J O'Connor and E F Robertson. Brook Taylor's Biography. [2009-12-25]. （原始内容存档于2010-11-20）.
2. ^ Rudin, 第123至124页.
3. ^ 《微积分(Ⅱ)》第88-90页.
4. ^ Klein (1998) 20.3; Apostol (1967) 7.7
5. ^ Protter, Morrey, 第135-136页