常用的麦克劳林级数展开式(泰勒展开式)
n = 0 , 1 , 2 , … 一般取到 x 的 3 ∼ 4 次方 n=0,1,2,\dots一般取到x的3\sim4次方 n=0,1,2,…一般取到x的3∼4次方
- e x = 1 + x 1 ! + x 2 2 ! + ⋯ + x n n ! + ο ( x n ) e^x=1+\displaystyle\frac{x}{1!}+\frac{x^2}{2!}+⋯+\frac{x^n}{n!}+\omicron(x^n) ex=1+1!x+2!x2+⋯+n!xn+ο(xn)
- ln ( 1 + x ) = x − x 2 2 + x 3 3 + ⋯ + ( − 1 ) n − 1 n x n + ο ( x n ) \ln{(1+x)}=x-\displaystyle\frac{x^2}{2}+\frac{x^3}{3}+⋯+\frac{(-1)^{n-1}}{n}x^n+\omicron(x^n) ln(1+x)=x−2x2+3x3+⋯+n(−1)n−1xn+ο(xn)
- sin x = x − x 3 3 ! + x 5 5 ! + ⋯ + ( − 1 ) n ( 2 n + 1 ) ! x 2 n + 1 + ο ( x 2 n + 1 ) \sin x=x-\displaystyle\frac{x^3}{3!}+\frac{x^5}{5!}+⋯+\frac{(-1)^{n}}{(2n+1)!}x^{2n+1}+\omicron(x^{2n+1}) sinx=x−3!x3+5!x5+⋯+(2n+1)!(−1)nx2n+1+ο(x2n+1)
- cos x = 1 − x 2 2 ! + x 4 4 ! + ⋯ + ( − 1 ) n ( 2 n ) ! x 2 n + ο ( x 2 n ) \cos x=1-\displaystyle\frac{x^2}{2!}+\frac{x^4}{4!}+⋯+\frac{(-1)^{n}}{(2n)!}x^{2n}+\omicron(x^{2n}) cosx=1−2!x2+4!x4+⋯+(2n)!(−1)nx2n+ο(x2n)
- t a n x = x + 1 3 x 3 + 1 5 x 5 + ⋯ + 1 2 n + 1 x 2 n + 1 + ο ( x 2 n + 1 ) tanx=x+\displaystyle\frac{1}{3}x^3+\frac{1}{5}x^5+\cdots+\frac{1}{2n+1}x^{2n+1}+\omicron(x^{2n+1}) tanx=x+31x3+51x5+⋯+2n+11x2n+1+ο(x2n+1)
推导： ( tan x − x ) ∼ 1 3 x 3 ∼ ( x − arctan x ) ( x − sin x ) ∼ 1 6 x 3 ∼ ( arcsin x − x ) α ∼ β ⇒ α = β + ο ( β ) 得 tan x = x + 1 3 x 3 + ο ( x 3 ) 同理 arctan x ， arcsin x \begin{aligned} 推导：&(\tan x -x)\sim\displaystyle\frac{1}{3}x^3\sim(x-\arctan x)\\ &(x−\sin x) \sim\displaystyle\frac{1}{6}x^3 \sim (\arcsin x−x)\\ &\alpha \sim \beta \Rightarrow \alpha=\beta+\omicron(\beta)\\ &得\ \tan x=x+\displaystyle\frac{1}{3}x^3+\omicron(x^3)\\ &同理\ \arctan x，\arcsin x &&&&&&&&&&&&&&&&&&&&&&&&&&&&& \end{aligned} 推导：(tanx−x)∼31x3∼(x−arctanx)(x−sinx)∼61x3∼(arcsinx−x)α∼β⇒α=β+ο(β)得 tanx=x+31x3+ο(x3)同理 arctanx，arcsinx - 1 1 − x = 1 + x + x 2 + ⋯ + ο ( x n ) \displaystyle\frac{1}{1-x}=1+x+x^2+\cdots+\omicron(x^n) 1−x1=1+x+x2+⋯+ο(xn)
- 1 1 + x = 1 − x + x 2 + ⋯ + ( − 1 ) n x n + ο ( x n ) \displaystyle\frac{1}{1+x}=1-x+x^2+\cdots+(-1)^nx^n+\omicron(x^n) 1+x1=1−x+x2+⋯+(−1)nxn+ο(xn)
- ( 1 + x ) a = 1 + a 1 ! x + a ( a − 1 ) 2 ! x 2 + ⋯ + a ( a − 1 ) ⋯ ( a − n + 1 ) n ! x n + ο ( x n ) (1+x)^a=1+\displaystyle\frac{a}{1!}x+\frac{a(a-1)}{2!}x^2+\cdots+\frac{a(a-1)\cdots(a-n+1)}{n!}x^n+\omicron(x^n) (1+x)a=1+1!ax+2!a(a−1)x2+⋯+n!a(a−1)⋯(a−n+1)xn+ο(xn)
- arcsin x = x + 1 2 × x 3 3 + 1 × 3 2 × 4 × x 5 5 + o ( x 5 ) = x + x 3 6 + ο ( x 3 ) \arcsin x=x+\displaystyle\frac{1}{2}\times\frac{x^3}{3}+\frac{1\times3}{2\times4}\times\frac{x^5}{5}+o(x^5)=x+\frac{x^3}{6}+\omicron(x^3) arcsinx=x+21×3x3+2×41×3×5x5+o(x5)=x+6x3+ο(x3)
- arctan x = x − x 3 3 + x 5 5 + ⋯ + ( − 1 ) n 2 n + 1 x 2 n + 1 + ο ( x 2 n + 1 ) \arctan x=x-\displaystyle\frac{x^3}{3}+\frac{x^5}{5}+⋯+\frac{(-1)^{n}}{2n+1}x^{2n+1}+\omicron(x^{2n+1}) arctanx=x−3x3+5x5+⋯+2n+1(−1)nx2n+1+ο(x2n+1)