This corresponds to a Riemann sum for the function f(x) = 1/(1 + x), over the closed interval [0, 1], using n subintervals of equal width. The graph below illustrates the lower Riemann sum on this interval for n = 8.
The Maclaurin series for ln(1 + x), valid for −1 < x 1, is:
The result now follows by setting x = 1.
More generally, if p and q are positive integers with p < q, then