令 g(x) = ln(1+x), g(0) = 0;
[ln(1+x)] ' = 1 / (1+x), g'(0) = 1;
[ln(1+x)] '' = -1 / (1+x)^2, g''(0) = -1;
[ln(1+x)] ''' = 2 / (1+x)^3, g''(0) = 2!; 一般有:
[ln(1+x)] ^(k) = (-1)^(k-1) * (k-1)! / (1+x)^k, g^(k)(0) = (-1)^(k-1) * (k-1)! ;
根据泰勒展开式有:
∴ ln(1+x) = x - x^2 / 2 + x^3 / 3 + ... ... + (-1)^(n-1) * x^n / n + ......
(1-x) * ln(1+x) = ln(1+x) - x * ln(1+x) = [x - x^2 / 2 + x^3 / 3 + ... ... + (-1)^(n-1) * x^n / n + .....] -
[x^2 - x^3 / 2 + x^4 / 3 + ... ... + (-1)^(n-1) * x^(n+1) / n + ......]
= x + (-1) * 3/2 * x^2 + 5/6 * x^3 + ... ... + (-1)^(n-1) * (2n-1) /[n * (n-1)] * x^n +.........
追答希望可以帮到你,大一时学过的