高等数学：泰勒公式
注：第三条 e x e^x ex的展开式，在 1 1 1和 + 1 2 x 2 +\frac{1}{2}x^2 +21x2之间添上一个 + x +x +x。
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1 1 − x = ∑ n = 0 ∞ x n = 1 + x + x 2 + x 3 + ο ( x 3 ) , x ∈ ( − 1 , 1 ) . \begin{aligned}\frac{1}{1-x}=\sum_{n=0}^\infty x^n=1+x+x^2+x^3+\omicron(x^3),x\in(-1,1).\end{aligned} 1−x1=n=0∑∞xn=1+x+x2+x3+ο(x3),x∈(−1,1).
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1 1 + x = ∑ n = 0 ∞ ( − 1 ) n x n = 1 − x + x 2 − x 3 + ο ( x 3 ) , x ∈ ( − 1 , 1 ) . \begin{aligned}\frac{1}{1+x}=\sum_{n=0}^\infty (-1)^nx^n=1-x+x^2-x^3+\omicron(x^3),x\in(-1,1).\end{aligned} 1+x1=n=0∑∞(−1)nxn=1−x+x2−x3+ο(x3),x∈(−1,1).
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e x = ∑ n = 0 ∞ x n n ! = 1 + 1 2 x 2 + 1 6 x 3 + ο ( x 3 ) , x ∈ ( − ∞ , + ∞ ) . \begin{aligned}e^x=\sum_{n=0}^\infty \frac{x^n}{n!}=1+\frac{1}{2}x^2+\frac{1}{6}x^3+\omicron(x^3),x\in(-\infty,+\infty).\end{aligned} ex=n=0∑∞n!xn=1+21x2+61x3+ο(x3),x∈(−∞,+∞).
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sin x = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! = x − x 3 6 + ο ( x 3 ) , x ∈ ( − ∞ , + ∞ ) . \begin{aligned}\sin x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{6}+\omicron(x^3),x\in(-\infty,+\infty).\end{aligned} sinx=n=0∑∞(−1)n(2n+1)!x2n+1=x−6x3+ο(x3),x∈(−∞,+∞).
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cos x = ∑ n = 0 ∞ ( − 1 ) n x 2 n ( 2 n ) ! = 1 − x 2 2 + x 4 24 + ο ( x 4 ) , x ∈ ( − ∞ , + ∞ ) . \begin{aligned}\cos x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2}+\frac{x^4}{24}+\omicron(x^4),x\in(-\infty,+\infty).\end{aligned} cosx=n=0∑∞(−1)n(2n)!x2n=1−2x2+24x4+ο(x4),x∈(−∞,+∞).
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tan x = ∑ n = 0 ∞ B 2 n ( − 4 ) n ( 1 − 4 n ) ( 2 n ) ! x 2 n − 1 = x + x 3 3 + ο ( x 3 ) , x ∈ ( − π 2 , π 2 ) . \begin{aligned}\tan x=\sum_{n=0}^\infty\frac{B_{2n}(-4)^n(1-4^n)}{(2n)!}x^{2n-1}=x+\frac{x^3}{3}+\omicron(x^3),x\in(-\frac{\pi}{2},\frac{\pi}{2}).\end{aligned} tanx=n=0∑∞(2n)!B2n(−4)n(1−4n)x2n−1=x+3x3+ο(x3),x∈(−2π,2π).
其中 B 2 n B_{2n} B2n 是 B e r n o u l l i \mathrm{Bernoulli} Bernoulli数，定义为 B n = lim x → 0 d n d x n [ x e x − 1 ] . \begin{aligned}B_n=\lim_{x\rightarrow0}\frac{d^n}{dx^n}[\frac{x}{e^x-1}].\end{aligned} Bn=x→0limdxndn[ex−1x].
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arcsin x = ∑ n = 0 ∞ ( 2 n ) ! 4 n ( n ! ) 2 × x 2 n + 1 2 n + 1 = x + x 3 6 + ο ( x 3 ) , x ∈ [ − 1 , 1 ] \begin{aligned}\arcsin x=\sum_{n=0}^\infty \frac{(2n)!}{4^n(n!)^2}\times\frac{x^{2n+1}}{2n+1}=x+\frac{x^3}{6}+\omicron(x^3),x\in[-1,1]\end{aligned} arcsinx=n=0∑∞4n(n!)2(2n)!×2n+1x2n+1=x+6x3+ο(x3),x∈[−1,1]
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arccos x = π 2 − arcsin x = π 2 − ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 2 n + 1 = π 2 − x − x 3 6 + ο ( x 3 ) , x ∈ [ − 1 , 1 ] . \begin{aligned}\arccos x=\frac{\pi}{2}-\arcsin x=\frac{\pi}{2}-\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{2n+1}=\frac{\pi}{2}-x-\frac{x^3}{6}+\omicron(x^3),x\in[-1,1].\end{aligned} arccosx=2π−arcsinx=2π−n=0∑∞2n+1(−1)nx2n+1=2π−x−6x3+ο(x3),x∈[−1,1].
注：一般的 T a y l o r Taylor Taylor公式表里面没有标注 arccos x \arccos x arccosx的原因是， arccos x + arcsin x = π 2 \arccos x+\arcsin x=\frac{\pi}{2} arccosx+arcsinx=2π，也就是说，根据 arcsin x \arcsin x arcsinx的 T a y l o r Taylor Taylor公式，就可以直接推出 $\arccos x 的 的 的Taylor$了。
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arctan x = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 2 n + 1 = x − x 3 3 + ο ( x 3 ) , x ∈ [ − 1 , 1 ] . \begin{aligned}\arctan x=\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{2n+1}=x-\frac{x^3}{3}+\omicron(x^3),x\in[-1,1].\end{aligned} arctanx=n=0∑∞2n+1(−1)nx2n+1=x−3x3+ο(x3),x∈[−1,1].
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a r c c o t x = π 2 − arctan x = π 2 − ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 2 n + 1 = π 2 − x + x 3 3 + ο ( x 3 ) , x ∈ [ − 1 , 1 ] . \begin{aligned}\mathrm{arccot} \,x=\frac{\pi}{2}-\arctan x=\frac{\pi}{2}-\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{2n+1}=\frac{\pi}{2}-x+\frac{x^3}{3}+\omicron(x^3),x\in[-1,1].\end{aligned} arccotx=2π−arctanx=2π−n=0∑∞2n+1(−1)nx2n+1=2π−x+3x3+ο(x3),x∈[−1,1].
这里也是一样，可以直接用 arctan x \arctan x arctanx 的 T a y l o r Taylor Taylor公式推出来，就不作过多解释了。
- a r c s e c x = arccos ( 1 x ) = π 2 − arcsin ( 1 x ) = π 2 − ∑ n = 0 ∞ ( 2 n ) ! 4 n ( n ! ) 2 × ( 1 x ) 2 n + 1 2 n + 1 = π 2 − 1 x − 1 6 x 3 + ο ( x 3 ) , { x ∈ R ∣ x ∉ ( − 1 , 1 ) } . \begin{aligned}\mathrm{arcsec}\,x=\arccos(\frac{1}{x})=\frac{\pi}{2}-\arcsin(\frac{1}{x})\end{aligned} \begin{aligned}=\frac{\pi}{2}-\sum_{n=0}^\infty\frac{(2n)!}{4^n(n!)^2}\times\frac{(\frac{1}{x})^{2n+1}}{2n+1}=\frac{\pi}{2}-\frac{1}{x}-\frac{1}{6x^3}+\omicron(x^3),\{x\in\mathbb{R}|x\notin(-1,1)\}.\end{aligned} arcsecx=arccos(x1)=2π−arcsin(x1)=2π−n=0∑∞4n(n!)2(2n)!×2n+1(x1)2n+1=2π−x1−6x31+ο(x3),{x∈R∣x∈/(−1,1)}.
至于怎么推导出来的，问就是desmos里图像完全一样。
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a r c c s c x = π 2 − a r c s e c x = π 2 − ( π 2 − arcsin ( 1 x ) ) = arcsin ( 1 x ) = ∑ n = 0 ∞ ( 2 n ) ! 4 n ( n ! ) 2 × ( 1 x ) 2 n + 1 2 n + 1 = 1 x + 1 6 x 3 + ο ( 1 x 3 ) , { x ∈ R ∣ x ∉ ( − 1 , 1 ) } . \begin{aligned}\mathbb{arccsc}\,x=\frac{\pi}{2}-\mathbb{arcsec}\,x=\frac{\pi}{2}-(\frac{\pi}{2}-\arcsin(\frac{1}{x}))=\arcsin(\frac{1}{x})\end{aligned}\begin{aligned}=\sum_{n=0}^\infty \frac{(2n)!}{4^n(n!)^2}\times\frac{(\frac{1}{x})^{2n+1}}{2n+1}=\frac{1}{x}+\frac{1}{6x^3}+\omicron(\frac{1}{x^3}),\{x\in\mathbb R|x\notin(-1,1)\}.\end{aligned} arccscx=2π−arcsecx=2π−(2π−arcsin(x1))=arcsin(x1)=n=0∑∞4n(n!)2(2n)!×2n+1(x1)2n+1=x1+6x31+ο(x31),{x∈R∣x∈/(−1,1)}.
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ln ( 1 + x ) = ∑ n = 0 ∞ ( − 1 ) n x n + 1 n + 1 = x − 1 2 x 2 + 1 3 x 3 + ο ( x 3 ) , x ∈ ( − 1 , 1 ] . \begin{aligned}\ln(1+x)=\sum_{n=0}^\infty(-1)^n\frac{x^{n+1}}{n+1}=x-\frac{1}{2}x^2+\frac{1}{3}x^3+\omicron(x^3),x\in(-1,1].\end{aligned} ln(1+x)=n=0∑∞(−1)nn+1xn+1=x−21x2+31x3+ο(x3),x∈(−1,1].
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( 1 + x ) m = 1 + ∑ n = 1 ∞ m ( m − 1 ) ⋯ ( m − n + 1 ) n ! x n , x ∈ ( − 1 , 1 ) . \begin{aligned}(1+x)^m=1+\sum_{n=1}^\infty\frac{m(m-1)\cdots(m-n+1)}{n!}x^n,x\in(-1,1).\end{aligned} (1+x)m=1+n=1∑∞n!m(m−1)⋯(m−n+1)xn,x∈(−1,1).
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cot x = ∑ n = 0 ∞ ( − 1 ) n 2 2 n B 2 n ( 2 n ) ! x 2 n − 1 = 1 x − 1 3 x − 1 45 x 3 + ο ( x 3 ) , x ∈ ( 0 , π ) . \begin{aligned}\cot x=\sum_{n=0}^\infty\frac{(-1)^n2^{2n}B_{2n}}{(2n)!}x^{2n-1}=\frac{1}{x}-\frac{1}{3}x-\frac{1}{45}x^3+\omicron(x^3),x\in(0,\pi).\end{aligned} cotx=n=0∑∞(2n)!(−1)n22nB2nx2n−1=x1−31x−451x3+ο(x3),x∈(0,π).
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sec x = ∑ n = 0 ∞ ( − 1 ) n E 2 n x 2 n ( 2 n ) ! = 1 + 1 2 x 2 + 5 24 x 4 + ο ( x 4 ) , x ∈ ( − π 2 , π 2 ) . \begin{aligned}\sec x=\sum_{n=0}^\infty\frac{(-1)^nE_{2n}x^{2n}}{(2n)!}=1+\frac{1}{2}x^2+\frac{5}{24}x^4+\omicron(x^4),x\in(-\frac{\pi}{2},\frac{\pi}{2}).\end{aligned} secx=n=0∑∞(2n)!(−1)nE2nx2n=1+21x2+245x4+ο(x4),x∈(−2π,2π).
其中 E 2 n E_{2n} E2n为 E u l e r Euler Euler数，定义为 E n = { 1 , n = 0. − ∑ k = 0 n − 1 ( − 1 ) k C 2 n 2 k E k , n ≥ 1. E_n= \begin{cases} 1,n=0.\\[2ex] \begin{aligned}-\sum_{k=0}^{n-1}\end{aligned}(-1)^kC_{2n}^{2k}E_k,n\ge1.\\[2ex] \end{cases} En=⎩ ⎨ ⎧1,n=0.−k=0∑n−1(−1)kC2n2kEk,n≥1.
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csc x = ∑ n = 0 ∞ ( − 1 ) n + 1 2 ( 2 n − 1 − 1 ) B 2 n ( 2 n ) ! x 2 n − 1 = 1 x + 1 6 x + 7 360 x 3 + ο ( x 3 ) , x ∈ ( 0 , π ) . \begin{aligned}\csc x=\sum_{n=0}^\infty\frac{(-1)^{n+1}2(2^{n-1}-1)B_{2n}}{(2n)!}x^{2n-1}=\frac{1}{x}+\frac{1}{6}x+\frac{7}{360}x^3+\omicron(x^3),x\in(0,\pi).\end{aligned} cscx=n=0∑∞(2n)!(−1)n+12(2n−1−1)B2nx2n−1=x1+61x+3607x3+ο(x3),x∈(0,π).
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sinh x = ∑ n = 0 ∞ x 2 n + 1 ( 2 n + 1 ) ! = x + 1 3 ! x 3 + ο ( x 3 ) , x ∈ ( − ∞ , + ∞ ) . \begin{aligned}\sinh x=\sum_{n=0}^\infty\frac{x^{2n+1}}{(2n+1)!}=x+\frac{1}{3!}x^3+\omicron(x^3),x\in(-\infty,+\infty).\end{aligned} sinhx=n=0∑∞(2n+1)!x2n+1=x+3!1x3+ο(x3),x∈(−∞,+∞).
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cosh x = ∑ n = 0 ∞ x 2 n ( 2 n ) ! = 1 + 1 2 ! x 2 + 1 4 ! x 4 + ο ( x 4 ) , x ∈ ( − ∞ , + ∞ ) . \begin{aligned}\cosh x=\sum_{n=0}^\infty\frac{x^{2n}}{(2n)!}=1+\frac{1}{2!}x^2+\frac{1}{4!}x^4+\omicron(x^4),x\in(-\infty,+\infty).\end{aligned} coshx=n=0∑∞(2n)!x2n=1+2!1x2+4!1x4+ο(x4),x∈(−∞,+∞).
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tanh x = ∑ n = 1 ∞ 2 2 n ( 2 2 n − 1 ) B 2 n x 2 n − 1 ( 2 n ) ! = x − 1 3 x 3 + ο ( x 3 ) , x ∈ ( − π 2 , π 2 ) . \begin{aligned}\tanh x=\sum_{n=1}^\infty\frac{2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}=x-\frac{1}{3}x^3+\omicron(x^3),x\in(-\frac{\pi}{2},\frac{\pi}{2}).\end{aligned} tanhx=n=1∑∞(2n)!22n(22n−1)B2nx2n−1=x−31x3+ο(x3),x∈(−2π,2π).
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coth x = ∑ n = 0 ∞ ( − 1 ) n − 1 2 2 n B n ( 2 n ! ) x 2 n − 1 = 1 x + 1 3 x − 1 45 x 3 + ο ( x 3 ) , x ∈ ( − π , π ) . \begin{aligned}\coth x=\sum_{n=0}^\infty\frac{(-1)^{n-1}2^{2n}B_{n}}{(2n!)}x^{2n-1}=\frac{1}{x}+\frac{1}{3}x-\frac{1}{45}x^3+\omicron(x^3),x\in(-\pi,\pi).\end{aligned} cothx=n=0∑∞(2n!)(−1)n−122nBnx2n−1=x1+31x−451x3+ο(x3),x∈(−π,π).
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s e c h x = ∑ n = 0 ∞ ( − 1 ) n E 2 n ( 2 n ) ! x 2 n = 1 − 1 2 ! x 2 + 5 4 ! x 4 + ο ( x 4 ) , x ∈ ( − π 2 , π 2 ) . \begin{aligned}\mathrm{sech}\,x=\sum_{n=0}^\infty\frac{(-1)^nE_{2n}}{(2n)!}x^{2n}=1-\frac{1}{2!}x^2+\frac{5}{4!}x^4+\omicron(x^4),x\in(-\frac{\pi}{2},\frac{\pi}{2}).\end{aligned} sechx=n=0∑∞(2n)!(−1)nE2nx2n=1−2!1x2+4!5x4+ο(x4),x∈(−2π,2π).
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c s c h x = ∑ n = 0 ∞ 2 ( − 1 ) n ( 2 2 n − 1 − 1 ) B n ( 2 n ) ! x 2 n − 1 = 1 x − 1 6 x + 7 360 x 3 + ο ( x 3 ) , x ∈ ( − π , π ) . \begin{aligned}\mathrm{csch}\,x=\sum_{n=0}^\infty\frac{2(-1)^n(2^{2n-1}-1)B_n}{(2n)!}x^{2n-1}=\frac{1}{x}-\frac{1}{6}x+\frac{7}{360}x^3+\omicron(x^3),x\in(-\pi,\pi).\end{aligned} cschx=n=0∑∞(2n)!2(−1)n(22n−1−1)Bnx2n−1=x1−61x+3607x3+ο(x3),x∈(−π,π).
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a r c s i n h x = ∑ n = 0 ∞ ( ( − 1 ) n ( 2 n ) ! 2 2 n ( n ! ) 2 ) x 2 n + 1 2 n + 1 = x − 1 6 x 3 + ο ( x 3 ) , x ∈ ( − 1 , 1 ) . \begin{aligned}\mathrm{arcsinh}\,x=\sum_{n=0}^\infty\begin{pmatrix}\frac{(-1)^n(2n)!}{2^{2n}(n!)^2}\end{pmatrix}\frac{x^{2n+1}}{2n+1}=x-\frac{1}{6}x^3+\omicron(x^3),x\in(-1,1).\end{aligned} arcsinhx=n=0∑∞(22n(n!)2(−1)n(2n)!)2n+1x2n+1=x−61x3+ο(x3),x∈(−1,1).
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a r c c o s h x = ln ( 2 x ) − ∑ n = 1 ∞ ( ( − 1 ) n ( 2 n ) ! 2 2 n ( n ! ) 2 ) x − 2 n 2 n = ln ( 2 x ) − 1 4 x − 2 − 3 32 x − 4 + ο ( x − 4 ) , { x ∈ R ∣ x ∉ [ − 1 , 1 ] } . \begin{aligned}\mathrm{arccosh}\,x=\ln(2x)-\sum_{n=1}^\infty\begin{pmatrix}\frac{(-1)^n(2n)!}{2^{2n}(n!)^2}\end{pmatrix}\frac{x^{-2n}}{2n}=\ln(2x)-\frac{1}{4}x^{-2}-\frac{3}{32}x^{-4}+\omicron(x^{-4}),\{x\in \mathbb{R}|x\notin[-1,1]\}.\end{aligned} arccoshx=ln(2x)−n=1∑∞(22n(n!)2(−1)n(2n)!)2nx−2n=ln(2x)−41x−2−323x−4+ο(x−4),{x∈R∣x∈/[−1,1]}.
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a r c t a n h x = ∑ n = 0 ∞ x 2 n + 1 2 n + 1 = x + 1 3 x 3 + ο ( x 3 ) , x ∈ ( − 1 , 1 ) . \begin{aligned}\mathrm{arctanh}\,x=\sum_{n=0}^\infty\frac{x^{2n+1}}{2n+1}=x+\frac{1}{3}x^3+\omicron(x^3),x\in(-1,1).\end{aligned} arctanhx=n=0∑∞2n+1x2n+1=x+31x3+ο(x3),x∈(−1,1).
a r c c o t h x \mathrm{arccoth}\,x arccothx 、 a r c s e c h x \mathrm{arcsech}\,x arcsechx 和 a r c c s c h x \mathrm{arccsch}\,x arccschx的公式找不到了。