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 +.}