微积分学/泰勒级数

泰勒级数

${\displaystyle \sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}}$

${\displaystyle \sin x}$ 及其1, 3, 5, 7, 9, 1113阶泰勒展开式的图像

原理

${\displaystyle f(x)={c_{0}}(x-a)^{0}+c_{1}(x-a)^{1}+c_{2}(x-a)^{2}+c_{3}(x-a)^{3}+\cdots +c_{n}(x-a)^{n}+\cdots }$

${\displaystyle \sum _{n=0}^{\infty }c_{n}(x-a)^{n}}$

${\displaystyle f(a)=c_{0}}$

${\displaystyle f'(x)=c_{1}(x-a)^{0}+2c_{2}(x-a)^{1}+3c_{3}(x-a)^{2}+4c_{4}(x-a)^{3}+\cdots +nc_{n}(x-a)^{(n-1)}+\cdots }$

${\displaystyle a}$ 代入得

${\displaystyle f'(a)=c_{1}}$

${\displaystyle f''(x)=2c_{2}+(2\times 3)c_{3}(x-a)^{1}+(3\times 4)c_{4}(x-a)^{2}+\cdots +(n)(n-1)c_{n}(x-a)^{(n-2)}+\cdots }$

${\displaystyle f''(a)=2c_{2}}$

${\displaystyle f'''(x)=(2\times 3)c_{3}(x-a)^{0}+(2\times 3\times 4)c_{4}(x-a)^{1}+(3\times 4\times 5)c_{5}(x-a)^{2}+\cdots +(n)(n-1)(n-2)c_{n}(x-a)^{n-3}}$

${\displaystyle f'''(a)=(2\times 3)c_{3}}$

${\displaystyle {\frac {d^{n}}{dx^{n}}}f(a)=n!\times c_{n}}$

${\displaystyle c_{n}={\frac {f^{(n)}(a)}{n!}}}$

${\displaystyle \sum _{n=0}^{\infty }c_{n}(x-a)^{n}}$

${\displaystyle f(x)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}}$

泰勒级数列表

${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}\quad {\text{对 任 意 }}x}$
${\displaystyle \ln(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}x^{n}\quad |x|<1}$

${\displaystyle {\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n}\quad |x|<1}$

${\displaystyle (1+x)^{\alpha }=\sum _{n=0}^{\infty }{\binom {\alpha }{n}}x^{n}\quad |x|<1}$

${\displaystyle \sin x=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{(2n-1)!}}x^{2n-1}}$
${\displaystyle \cos x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}}$
${\displaystyle \tan x=\sum _{n=1}^{\infty }{\frac {B_{2n}(-4)^{n}(1-4^{n})}{(2n)!}}x^{2n-1}\quad |x|<{\frac {\pi }{2}}}$
${\displaystyle \sec x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}}{(2n)!}}x^{2n}\quad |x|<{\frac {\pi }{2}}}$
${\displaystyle \arcsin x=\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}\quad |x|<1}$
${\displaystyle \arctan x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{2n+1}\quad |x|<1}$

${\displaystyle \sinh x=\sum _{n=1}^{\infty }{\frac {x^{2n-1}}{(2n-1)!}}\quad {\text{对 任 意 }}x}$
${\displaystyle \cosh x=\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}\quad {\text{对 任 意 }}x}$
${\displaystyle \tanh x=\sum _{n=1}^{\infty }{\frac {B_{2n}2^{2n}(2^{2n}-1)}{(2n)!}}x^{2n-1}\quad |x|<{\frac {\pi }{2}}}$
${\displaystyle {\rm {arsinh}}x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}(2n+1)}}x^{2n+1}\quad |x|<1}$
${\displaystyle {\rm {artanh}}x=\sum _{n=1}^{\infty }{\frac {x^{2n-1}}{2n-1}}\quad |x|<1}$

${\displaystyle W_{0}(x)=\sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}x^{n}\quad |x|<{\frac {1}{e}}}$

例题

例1

${\displaystyle f(x)=\ln {\big (}1+\cos x{\big )}}$

解答

${\displaystyle \ln(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}x^{n}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots \quad |x|<1}$

${\displaystyle \cos x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots \quad {\text{对 任 意 }}x\in \mathbb {C} }$

${\displaystyle \left(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots \right)-{\frac {1}{2}}\left(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots \right)^{2}+{\frac {1}{3}}\left(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots \right)^{3}-\cdots }$

${\displaystyle \ln {\big (}1+\cos x{\big )}=\ln 2-{\frac {x^{2}}{4}}-{\frac {x^{4}}{96}}-{\frac {x^{6}}{1440}}-{\frac {17x^{8}}{322560}}-{\frac {31x^{10}}{7257600}}-\cdots }$

例2

${\displaystyle g(x)={\frac {e^{x}}{\cos x}}}$

解答

${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots }$

${\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots }$

${\displaystyle {\frac {e^{x}}{\cos x}}=c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+\cdots }$

 ${\displaystyle e^{x}}$ ${\displaystyle ={\big (}c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+\cdots {\big )}\cos x}$ ${\displaystyle =\left(c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+c_{4}x^{4}+\cdots \right)\left(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots \right)}$ ${\displaystyle =c_{0}-{\frac {c_{0}}{2}}x^{2}+{\frac {c_{0}}{4!}}x^{4}+c_{1}x-{\frac {c_{1}}{2}}x^{3}+{\frac {c_{1}}{4!}}x^{5}+c_{2}x^{2}-{\frac {c_{2}}{2}}x^{4}+{\frac {c_{2}}{4!}}x^{6}+c_{3}x^{3}-{\frac {c_{3}}{2}}x^{5}+{\frac {c_{3}}{4!}}x^{7}+\cdots }$

${\displaystyle e^{x}=c_{0}+c_{1}x+\left(c_{2}-{\frac {c_{0}}{2}}\right)x^{2}+\left(c_{3}-{\frac {c_{1}}{2}}\right)x^{3}+\left(c_{4}+{\frac {c_{0}}{4!}}-{\frac {c_{2}}{2}}\right)x^{4}+\cdots }$

${\displaystyle {\frac {e^{x}}{\cos x}}=1+x+x^{2}+{\frac {2x^{3}}{3}}+{\frac {x^{4}}{2}}+\cdots }$