【考研数学高数部分】泰勒展开式

泰勒展开式

f(x)=f(x0)+f′(x0)(x−x0)+f′′(x0)2!(x−x0)2+...+f(n)(x0)n!(x−x0)n+f(n+1)(x0+θ(x−x0))(n+1)!(x−x0)n+1(0<θ<1)意义：可用n次多项式来近似表达函数f(x), 且误差是当x→x0时比(x−x0)n高阶无穷小\begin{array}{l} \\ f(x) = f(x_0)+f'(x_0)(x-x_0)\\ \\ +{\frac{f''(x_0)}{2!}}{(x-x_0)^2}+...\\ \\ +{\frac{f^{(n)}(x_0)}{n!}}{(x-x_0)^n}\\ \\ +{\frac{f^{(n+1)}(x_0+\theta(x-x_0))}{(n+1)!}}{(x-x_0)^{n+1}} (0<\theta<1)\\ \end{array}\\ {\text{意义：可用n次多项式来近似表达函数f(x), 且误差是当}}x\rightarrow x_0 {\text{ 时比}}(x-x_0)^n{\text{高阶无穷小}} f(x)=f(x0)+f′(x0)(x−x0)+2!f′′(x0)(x−x0)2+...+n!f(n)(x0)(x−x0)n+(n+1)!f(n+1)(x0+θ(x−x0))(x−x0)n+1(0<θ<1)意义：可用n次多项式来近似表达函数f(x), 且误差是当x→x0 时比(x−x0)n高阶无穷小

一些常用的泰勒展开式：
sinx=x−13!x3+15!x5+⋯+(−1)nx2n+1(2n+1)!=∑n=0∞(−1)nx2n+1(2n+1)!cosx=1−12!x2+14!x4+⋯+(−1)nx2n(2n)!=∑n=0∞(−1)nx2n(2n)!arcsinx=x+13!x3+o(x3)tanx=x+13x3+o(x3)arctanx=x−13x3+o(x3)ln(1+x)=x−12x2+13x3−⋯+(−1)n−1xnn=∑n=1∞(−1)n−1xnn(−1<x≤1)ex=1+x1!+x22!+x33!+⋯+xnn!=∑n=0∞xnn!(1+x)a=1+a1!x+a(a−1)2!x2+a(a−1)(a−2)3!x3+o(x3)11−x=1+x+x2+x3+⋯+xn=∑n=0∞xn(−1<x<1)11+x=1−x+x2−x3+⋯+(−1)nxn=∑n=0∞(−1)nxn(−1<x<1)ax=eln(ax)=e(xlna)=∑(xlna)nn!ax−1=xlnasinx = x - {\frac{1}{3!}}x^3+{\frac{1}{5!}}x^5+\dots+(-1)^n\frac{x^{2n+1}}{(2n+1)!}=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}\\ cosx = 1 - {\frac{1}{2!}}x^2+{\frac{1}{4!}}x^4+\dots+(-1)^n\frac{x^{2n}}{(2n)!}=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}\\ arcsinx = x + {\frac{1}{3!}}x^3+o(x^3)\\ tanx = x + {\frac{1}{3}}x^3+o(x^3)\\ arctanx = x - {\frac{1}{3}}x^3+o(x^3)\\ ln(1+x) = x-{\frac{1}{2}}x^2+{\frac{1}{3}}x^3-\dots+(-1)^{n-1}\frac{x^n}{n}=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n} \ \ \ \ (-1<x\le 1)\\ e^x = 1+{\frac{x}{1!}}+{\frac{x^2}{2!}}+{\frac{x^3}{3!}}+\dots+\frac{x^n}{n!}=\sum_{n=0}^{\infty}\frac{x^n}{n!}\\ (1+x)^a = 1+{\frac{a}{1!}}x+{\frac{a(a-1)}{2!}}x^2+{\frac{a(a-1)(a-2)}{3!}}x^3+o(x^3)\\ {\frac{1}{1-x}} = 1 + x+x^2+x^3+\dots+x^n=\sum_{n=0}^{\infty}x^n \ \ \ \ (-1<x<1)\\ {\frac{1}{1+x}} = 1 - x + x^2 - x^3+\dots+(-1)^nx^n=\sum_{n=0}^{\infty}(-1)^nx^n \ \ \ \ (-1<x<1)\\ \\ a^x=e^{ln(a^x)}=e^{(xlna)} =∑\frac{(xlna)^n}{n!} a^x-1=xlna\\ sinx=x−3!1x3+5!1x5+⋯+(−1)n(2n+1)!x2n+1=n=0∑∞(−1)n(2n+1)!x2n+1cosx=1−2!1x2+4!1x4+⋯+(−1)n(2n)!x2n=n=0∑∞(−1)n(2n)!x2narcsinx=x+3!1x3+o(x3)tanx=x+31x3+o(x3)arctanx=x−31x3+o(x3)ln(1+x)=x−21x2+31x3−⋯+(−1)n−1nxn=n=1∑∞(−1)n−1nxn (−1<x≤1)ex=1+1!x+2!x2+3!x3+⋯+n!xn=n=0∑∞n!xn(1+x)a=1+1!ax+2!a(a−1)x2+3!a(a−1)(a−2)x3+o(x3)1−x1=1+x+x2+x3+⋯+xn=n=0∑∞xn (−1<x<1)1+x1=1−x+x2−x3+⋯+(−1)nxn=n=0∑∞(−1)nxn (−1<x<1)ax=eln(ax)=e(xlna)=∑n!(xlna)nax−1=xlna

等价无穷小

lnx=ln(1+x−1)∼x−1lnu∼u−1,(u→0)lnx=ln(1+x-1) \sim x-1\\ lnu\sim u-1,(u\to 0) lnx=ln(1+x−1)∼x−1lnu∼u−1,(u→0)

高等数学中的
β=O(α)表示β是α的同阶无穷小
β=o(α)表示β是α的高阶无穷小