查看“泰勒公式”的源代码

== 泰勒公式 ==
如果函数<math>f\left ( x \right )</math>在<math>x=x_{0} </math>的领域内具有n+1阶导数则
<math>f\left(x\right) =f\left( x_{0} \right)+f'\left ( x_{0}\right)\left(x-x_{0}\right )+\cdots+\frac{f^{\left(n\right)}\left( x_{0}\right )}{n!}\left(x-x_{0}\right )^{n}+R_{n}\left(x\right ) </math>

其中<math>\xi </math>介于<math>x</math>与<math>x_{0} </math>之间，<math>R_{n}\left(x\right )  =\frac{f^{\left(n+1\right)}\left( \xi \right)}{\left(n+1\right)!}\left(x-x_{0}\right)^{n+1}</math>称之为拉格朗日余项，余项<math>R_{n}\left(x\right)</math>也可以表示为<math>R_{n}\left(x\right)=o\left(\left(x-x_{0}\right)^{n}\right )</math>。

（1）当<math>x_{0}=0</math>时，<math>f\left ( x \right ) =f\left ( 0 \right ) +f^{'} \left ( 0 \right ) x+\cdots +\frac{f^{\left(n\right)}\left(0\right)}{n!}\left(x\right)^{n}+R_{n} \left ( x \right ) </math>称为麦克劳林公式。

（2）常用的麦克劳林公式：

<math>\ \ \ \ </math><big>①</big><math>\ \ e^{x} =1+x+\frac{x^{2} }{2!} +\cdots +\frac{x^{n} }{n!} + o \left ( x^{n}  \right ) </math>

<math>\ \ \ \ </math><big>②</big><math>\ \ \sin x=x-\frac{x^{3} }{3!}+\cdots +\frac{\left ( -1 \right )^{n}  }{\left ( 2n+1 \right )! } x^{2n+1} + o \left ( x^{2n+1}  \right ) </math>

<math>\ \ \ \ </math><big>③</big><math>\ \ \cos x=1-\frac{x^{2} }{2!} +\cdots +\frac{\left ( -1 \right )^{n}  }{\left ( 2n \right )! } x^{2n}+ o \left ( x^{2n}  \right )   </math>

<math>\ \ \ \ </math><big>④</big><math>\ \ \frac{1}{1-x} =1+x+x^{2} +\cdots +x^{n} + o \left ( x^{n}  \right )  </math>

<math>\ \ \ \ </math><big>⑤</big><math>\ \ \frac{1}{1+x}=1-x+x^{2} -\cdots +\left ( -1 \right ) ^{n}x^{n}  + o \left ( x^{n}  \right ) </math>

<math>\ \ \ \ </math><big>⑥</big><math>\ \ \ln_{}{\left ( 1+x \right ) } =x-\frac{x^{2} }{2} +\frac{x^{3} }{3} -\cdots +\frac{\left ( -1 \right ) ^{n-1} }{n}x^{n} + o \left ( x^{n}  \right ) </math>

<math>\ \ \ \ </math><big>⑦</big><math>\ \ \left ( 1+x \right )^{a}  =1+ax+\frac{a\left ( a-1 \right ) }{2!}x^{2}+\cdots +\frac{a\left(a-1\right)\cdots \left(a-n+1\right)}{n!}x^{n}  + o \left ( x^{n}  \right )</math>

<math>\ \ \ \ </math><big>⑧</big><math>\ \ \arctan x=x-\frac{x^{3} }{3} +\frac{x^{5} }{5} -\cdots +\frac{\left ( -1 \right )^{n}}{2n+1} x^{2n+1}+ o \left ( x^{2n+1}  \right ) </math>