Within the radius of convergence, the power series of a function fits closely to the function itself. For ln(1+X), where X = (x-1), the power series is centered on 1, and the radius of convergence of the power series is 1. The series converges on the interval -1 < X ≤ 1. We have plotted the function and the power series on -1.2 < X ≤ 1.2 (or -.2 < x ≤ 2.2). The figure illustrates both the closeness of the fit and the divergence outside the interval after 30 terms. The figure is based on Needham's Chapter 2, Exercise 31, p. 119.
The power series for ln(1+X) is found by integrating the terms of the power series for 1/(1+x) on the interval [0, X]. The power series for 1/(1+x) is found with the binomial theorem.