A function and its n^th degree Taylor polynomial

\textrm{Based on the following MacLaurin polynomial approximations} \\ \\ \mbox{}\qquad \ln(1 + x) \approx x - \frac{1}{2}x^2 + \frac{1}{3} x^3 - \frac{1}{4}x^4 + \frac{1}{5}x^5 - \cdots + (-1)^{n-1}\frac{1}{n}x^n, \\ \mbox{}\qquad \ln(1 - x) \approx -x - \frac{1}{2}x^2 - \frac{1}{3} x^3 - \frac{1}{4}x^4 - \frac{1}{5}x^5 - \cdots - \frac{1}{n}x^n, \\ \\ \textrm{we subtract to obtain the approximation} \\ \\ \mbox{}\qquad\ln\left(\frac{1+x}{1-x}\right) \approx 2\left(x + \frac{1}{3}x^3 + \frac{1}{5}x^5 + \cdots + \frac{1}{2k+1}x^{2k+1}\right) = T(x), \\ \\ \textrm{the } (2k+1)-\textrm{degree MacLaurin polynomial of } \ln((1+x)/(1-x)). \\