考研中出现过的泰勒展开式

汇总：

 

1. e^x的泰勒展开式：e^x = 1 + x + x^2/2! + x^3/3! + ... + x^n/n! + ...
2. sinx的泰勒展开式：sinx = x - x^3/3! + x^5/5! - x^7/7! + ... + (-1)^n*x^(2n+1)/(2n+1)! + ...
3. cosx的泰勒展开式：cosx = 1 - x^2/2! + x^4/4! - x^6/6! + ... + (-1)^n*x^(2n)/(2n)! + ...
4. ln(1+x)的泰勒展开式：ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ... + (-1)^(n-1)*x^n/n + ...
5. tanx的泰勒展开式：tanx = x + x^3/3 + 2x^5/15 + 17x^7/315 + ... + Bernoulli(2n)2^(2n)(2^(2n)-1)*B(2n)*x^(2n-1)/(2n)! + ...
6. arctanx的泰勒展开式：arctanx = x - x^3/3 + x^5/5 - x^7/7 + ... + (-1)^(n-1)*x^(2n-1)/(2n-1) + ...
7. e^(-x^2/2)的泰勒展开式：e^(-x^2/2) = 1 - x^2/2 + x^4/8 - x^6/48 + ... + (-1)^nx^(2n)/(2^nn!) + ...
8. 1/(1-x)的泰勒展开式：1/(1-x) = 1 + x + x^2 + x^3 + ... + x^n + ...
9. (1+x)^m的泰勒展开式：(1+x)^m = 1 + mx + m(m-1)x^2/2! + m(m-1)(m-2)x^3/3! + ... + m(m-1)...*(m-n+1)*x^n/n! + ...
10. sqrt(1+x)的泰勒展开式（麦克劳林展开式）：sqrt(1+x) = 1 + x/2 - x^2/8 + x^3/16 - x^4/128 + ... + (-1)^(n-1)*(2n-1)!!x^n/(2^nn!) + ...
11. f(x+a)的泰勒展开式：f(x+a) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + f^(n)(a)*(x-a)^n/n! + R_n(x)
12. ln(x+1)的泰勒展开式：ln(x+1) = x - x^2/2 + x^3/3 - x^4/4 + ... + (-1)^(n-1)*x^n/n + R_n(x)
13. sinhx的泰勒展开式：sinhx = x + x^3/3! + x^5/5! + ... + x^(2n+1)/(2n+1)! + ...
14. coshx的泰勒展开式：coshx = 1 + x^2/2! + x^4/4! + ... + x^(2n)/(2n)! + ...
15. (1+x)^a的泰勒展开式：(1+x)^a = 1 + ax + a(a-1)x^2/2! + a(a-1)(a-2)x^3/3! + ... + a(a-1)...*(a-n+1)*x^n/n! + R_n(x)
16. ln(1-x)的泰勒展开式：ln(1-x) = -x - x^2/2 - x^3/3 - ... - x^n/n + R_n(x), |x|<1