8个常见的泰勒公式

8个常用泰勒公式:

sin ⁡ x = x − 1 6 x 3 + O ( x 3 ) arcsin ⁡ x = x + 1 6 x 3 + O ( x 3 ) sin x=x-frac{1}{6} x^{3}+Oleft(x^{3}

ight) quad arcsin x=x+frac{1}{6} x^{3}+Oleft(x^{3}

ight)sinx=x−

6

1

x

3

+O(x

3

)arcsinx=x+

6

1

x

3

+O(x

3

)

cos ⁡ x = 1 − 1 2 x 2 + x 4 4 ! + 0 ( x 4 ) ln ⁡ ( 1 + x ) = x − 1 2 x 2 + 1 3 x 3 + O ( x 3 ) cos x=1-frac{1}{2} x^{2}+frac{x^{4}}{4 !}+0left(x^{4}

ight) quad ln (1+x)=x-frac{1}{2} x^{2}+frac{1}{3} x^{3}+O(x^{3})cosx=1−

2

1

x

2

+

4!

x

4

+0(x

4

)ln(1+x)=x−

2

1

x

2

+

3

1

x

3

+O(x

3

)

tan ⁡ x = x + 1 3 x 3 + O ( x 3 ) arctan ⁡ x = x − 1 3 x 3 + O ( x 3 )

an x=x+frac{1}{3} x^{3}+O( x^{3}) quad arctan x=x-frac{1}{3} x^{3}+Oleft(x^{3}

ight)tanx=x+

3

1

x

3

+O(x

3

)arctanx=x−

3

1

x

3

+O(x

3

)

e x = 1 + x + 1 2 x 2 + 1 6 x 3 + 0 ( x 3 ) ( 1 + x ) a = 1 + a x + + a ( a − 1 ) 2 ! x 2 + O ( x 2 ) e^{x}=1+x+frac{1}{2} x^{2}+frac{1}{6} x^{3}+0left(x^{3}

ight) quad(1+x)^{a}=1+a x++frac{a(a-1)}{2 !} x^{2}+Oleft(x^{2}

ight)e

x

=1+x+

2

1

x

2

+

6

1

x

3

+0(x

3

)(1+x)

a

=1+ax++

2!

a(a−1)

x

2

+O(x

2

)

泰勒公式是等号而不是等价，这就使所有函数转化为幂函数，在利用高阶无穷小被低阶吸收的原理，可以秒杀大部分极限题。

1、e^x = 1+x+x^2/2!+x^3/3!+……+x^n!+……

2、ln(1+x)=x-x^2/2+x^3/3-……+(-1)^(k-1)*(x^k)/k(|x|<1)

3、sin x = x-x^3/3!+x^5/5!-……+(-1)^(k-1)*(x^(2k-1))/(2k-1)!+……。(-∞<x<∞)

4、cos x = 1-x^2/2!+x^4/4!-……+(-1)k*(x^(2k))/(2k)!+…… (-∞<x<∞)

5、arcsin x = x + 1/2*x^3/3 + 1*3/(2*4)*x^5/5 + ……(|x|<1)

6、arccos x = π - ( x + 1/2*x^3/3 + 1*3/(2*4)*x^5/5 + …… ) (|x|<1)

7、arctan x = x - x^3/3 + x^5/5 -……(x≤1)

8、sinh x = x+x^3/3!+x^5/5!+……+(-1)^(k-1)*(x^2k-1)/(2k-1)!+……