Sequences and Series, The Harmonic Series

The Harmonic Series

The series hn = 1/n is called the harmonic series, since the terms represent the harmonics of a fundamental wave form. This is an example of a divergent series whose terms approach 0. To prove this, we will show that h dominates an unbounded series. Define the series g as:

1 1/2 1/4 1/4 1/8 1/8 1/8 1/8 1/16 (8 of these) 1/32 (16 of these) …

The second term is 1/2, and the third and fourth terms add up to 1/2. The next four terms add up to 1/2, the next 8 terms add up to 1/2, the next 16 terms add up to 1/2, and so on. We keep adding halves forever, hence g is unbounded. Since h dominates g, h is unbounded.

Let r be the sequence of partial sums of h. Recall that the log of n is the area under the curve y = 1/x from 1 to n. Now rn-1 is a set of regularly spaced rectangles approximating the area of the curve. Shifting the rectangles to the left one unit, rn-1 also approximates the area. The former is too large (upper sums), while the latter is too small (lower sums). We thus have rn-1 < log(n) < rn-1. Note that log(n) is bracketed between two numbers that are never more than 1 apart. Thus the difference between rn and log(n) is no more than 1.