Perhaps the most interesting and useful fact about the harmonic series is that the nth harmonic number (the partial sum of the first n terms) is an extremely ac

Perhaps the most interesting and useful fact about the harmonic series is that the nth harmonic number (the partial sum of the first n terms) is an extremely accurately approximation of ln(n) (they are very close for all values of n, in the limit they differ only by the Euler-Mascheroni constant, which is about 0.577). This is the only approximation I have seen that has constant error in the limit.