泰勒展开式
f ( x ) = f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) + f ′ ′ ( x 0 ) 2 ! ( x − x 0 ) 2 + . . . + f ( n ) ( x 0 ) n ! ( x − x 0 ) n + f ( n + 1 ) ( x 0 + θ ( x − x 0 ) ) ( n + 1 ) ! ( x − x 0 ) 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 → x 0 时比 ( x − x 0 ) 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：
s i n x = x − 1 3 ! x 3 + 1 5 ! x 5 + ⋯ + ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! c o s x = 1 − 1 2 ! x 2 + 1 4 ! x 4 + ⋯ + ( − 1 ) n x 2 n ( 2 n ) ! = ∑ n = 0 ∞ ( − 1 ) n x 2 n ( 2 n ) ! a r c s i n x = x + 1 3 ! x 3 + o ( x 3 ) t a n x = x + 1 3 x 3 + o ( x 3 ) a r c t a n x = x − 1 3 x 3 + o ( x 3 ) l n ( 1 + x ) = x − 1 2 x 2 + 1 3 x 3 − ⋯ + ( − 1 ) n − 1 x n n = ∑ n = 1 ∞ ( − 1 ) n − 1 x n n ( − 1 < x ≤ 1 ) e x = 1 + x 1 ! + x 2 2 ! + x 3 3 ! + ⋯ + x n n ! = ∑ n = 0 ∞ x n n ! ( 1 + x ) a = 1 + a 1 ! x + a ( a − 1 ) 2 ! x 2 + a ( a − 1 ) ( a − 2 ) 3 ! x 3 + o ( x 3 ) 1 1 − x = 1 + x + x 2 + x 3 + ⋯ + x n = ∑ n = 0 ∞ x n ( − 1 < x < 1 ) 1 1 + x = 1 − x + x 2 − x 3 + ⋯ + ( − 1 ) n x n = ∑ n = 0 ∞ ( − 1 ) n x n ( − 1 < x < 1 ) a x = e l n ( a x ) = e ( x l n a ) = ∑ ( x l n a ) n n ! a x − 1 = x l n a sinx = 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
l n x = l n ( 1 + x − 1 ) ∼ x − 1 l n u ∼ 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