高等数学：泰勒公式

注：第三条$e^x$的展开式，在$1$和$+\frac{1}{2}{x}^2$之间添上一个$+x$。

1. $\begin{array}{r}\frac{1}{1-x}=\sum _{n=0}^{\infty }{x}^n=1+x+x^2+x^3+o(x^3),x\in (-1,1).\end{array}$

2. $\begin{array}{r}\frac{1}{1+x}=\sum _{n=0}^{\infty }(-1{)}^n{x}^n=1-x+x^2-x^3+o(x^3),x\in (-1,1).\end{array}$

3. $\begin{array}{r}{e}^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}=1+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+o(x^3),x\in (-\infty ,+\infty ).\end{array}$

4. $\begin{array}{r}\sin x=\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{6}+o(x^3),x\in (-\infty ,+\infty ).\end{array}$

5. $\begin{array}{r}\cos x=\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2}+\frac{x^4}{24}+o(x^4),x\in (-\infty ,+\infty ).\end{array}$

6. $\begin{array}{r}\tan x=\sum _{n=0}^{\infty }\frac{B_{2n}(-4{)}^n(1-4^n)}{(2n)!}{x}^{2n-1}=x+\frac{x^3}{3}+o(x^3),x\in (-\frac{\pi }{2},\frac{\pi }{2}).\end{array}$

其中 $B_{2n}$ 是 $\mathrm{Bernoulli}$数，定义为 $\begin{array}{r}{B}_n=\underset{x\to 0}{\lim }\frac{d^n}{d{x}^n}[\frac{x}{e^x-1}].\end{array}$

7. $\begin{array}{r}\arcsin x=\sum _{n=0}^{\infty }\frac{(2n)!}{4^n(n!{)}^2}\times \frac{x^{2n+1}}{2n+1}=x+\frac{x^3}{6}+o(x^3),x\in [-1,1]\end{array}$

8. $\begin{array}{r}\arccos x=\frac{\pi }{2}-\arcsin x=\frac{\pi }{2}-\sum _{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n+1}}{2n+1}=\frac{\pi }{2}-x-\frac{x^3}{6}+o(x^3),x\in [-1,1].\end{array}$

注：一般的$Taylor$公式表里面没有标注 $\arccos x$的原因是， $\arccos x+\arcsin x=\frac{\pi }{2}$，也就是说，根据 $\arcsin x$的$Taylor$公式，就可以直接推出 $\arccos x $的$Taylor$了。

9. $\begin{array}{r}\arctan x=\sum _{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n+1}}{2n+1}=x-\frac{x^3}{3}+o(x^3),x\in [-1,1].\end{array}$

10. $\begin{array}{r}\mathrm{arccot}x=\frac{\pi }{2}-\arctan x=\frac{\pi }{2}-\sum _{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n+1}}{2n+1}=\frac{\pi }{2}-x+\frac{x^3}{3}+o(x^3),x\in [-1,1].\end{array}$

这里也是一样，可以直接用 $\arctan x$ 的$Taylor$公式推出来，就不作过多解释了。

11. 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。

12. 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)}.​

13. $\begin{array}{r}\ln (1+x)=\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{n+1}}{n+1}=x-\frac{1}{2}{x}^2+\frac{1}{3}{x}^3+o(x^3),x\in (-1,1].\end{array}$

14. $\begin{array}{r}(1+x{)}^m=1+\sum _{n=1}^{\infty }\frac{m(m-1)\cdots (m-n+1)}{n!}{x}^n,x\in (-1,1).\end{array}$

15. $\begin{array}{r}\cot x=\sum _{n=0}^{\infty }\frac{(-1{)}^n{2}^{2n}{B}_{2n}}{(2n)!}{x}^{2n-1}=\frac{1}{x}-\frac{1}{3}x-\frac{1}{45}{x}^3+o(x^3),x\in (0,\pi ).\end{array}$

16. $\begin{array}{r}\sec x=\sum _{n=0}^{\infty }\frac{(-1{)}^n{E}_{2n}{x}^{2n}}{(2n)!}=1+\frac{1}{2}{x}^2+\frac{5}{24}{x}^4+o(x^4),x\in (-\frac{\pi }{2},\frac{\pi }{2}).\end{array}$

其中 $E_{2n}$为$Euler$数，定义为 $E_n=\{\begin{array}{l}1,n=0.\\ \begin{array}{r}-\sum _{k=0}^{n-1}\end{array}(-1{)}^k{C}_{2n}^{2k}{E}_k,n\ge 1.\end{array}$

17. $\begin{array}{r}\csc x=\sum _{n=0}^{\infty }\frac{(-1{)}^{n+1}2(2^{n-1}-1)B_{2n}}{(2n)!}{x}^{2n-1}=\frac{1}{x}+\frac{1}{6}x+\frac{7}{360}{x}^3+o(x^3),x\in (0,\pi ).\end{array}$

18. $\begin{array}{r}\sinh x=\sum _{n=0}^{\infty }\frac{x^{2n+1}}{(2n+1)!}=x+\frac{1}{3!}{x}^3+o(x^3),x\in (-\infty ,+\infty ).\end{array}$

19. $\begin{array}{r}\cosh x=\sum _{n=0}^{\infty }\frac{x^{2n}}{(2n)!}=1+\frac{1}{2!}{x}^2+\frac{1}{4!}{x}^4+o(x^4),x\in (-\infty ,+\infty ).\end{array}$

20. $\begin{array}{r}\tanh x=\sum _{n=1}^{\infty }\frac{2^{2n}(2^{2n}-1)B_{2n}{x}^{2n-1}}{(2n)!}=x-\frac{1}{3}{x}^3+o(x^3),x\in (-\frac{\pi }{2},\frac{\pi }{2}).\end{array}$

21. $\begin{array}{r}\coth x=\sum _{n=0}^{\infty }\frac{(-1{)}^{n-1}{2}^{2n}{B}_n}{(2n!)}{x}^{2n-1}=\frac{1}{x}+\frac{1}{3}x-\frac{1}{45}{x}^3+o(x^3),x\in (-\pi ,\pi ).\end{array}$

22. $\begin{array}{r}\mathrm{sech}x=\sum _{n=0}^{\infty }\frac{(-1{)}^n{E}_{2n}}{(2n)!}{x}^{2n}=1-\frac{1}{2!}{x}^2+\frac{5}{4!}{x}^4+o(x^4),x\in (-\frac{\pi }{2},\frac{\pi }{2}).\end{array}$

23. $\begin{array}{r}\mathrm{csch}x=\sum _{n=0}^{\infty }\frac{2(-1{)}^n(2^{2n-1}-1)B_n}{(2n)!}{x}^{2n-1}=\frac{1}{x}-\frac{1}{6}x+\frac{7}{360}{x}^3+o(x^3),x\in (-\pi ,\pi ).\end{array}$

24. $\begin{array}{r}\mathrm{arcsinh}x=\sum _{n=0}^{\infty }\left(\begin{array}{c}\frac{(-1{)}^n(2n)!}{2^{2n}(n!{)}^2}\end{array}\right)\frac{x^{2n+1}}{2n+1}=x-\frac{1}{6}{x}^3+o(x^3),x\in (-1,1).\end{array}$

25. 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)−41​x−2−323​x−4+ο(x−4),{x∈R∣x∈/[−1,1]}.​

26. $\begin{array}{r}\mathrm{arctanh}x=\sum _{n=0}^{\infty }\frac{x^{2n+1}}{2n+1}=x+\frac{1}{3}{x}^3+o(x^3),x\in (-1,1).\end{array}$

$\mathrm{arccoth}x$ 、 $\mathrm{arcsech}x$ 和 $\mathrm{arccsch}x$的公式找不到了。