Consider the numbers 1, then 1 + 1/2, then 1 + (1/2) + (1/3), then 1 + (1/2) + (1/3) + (1/4), and so on. These are called the harmonic numbers, denoted Hn. Thus H1 = 1, H2 = 3/2, and H3 = 11/6, and so on.
Of course, the farther along the sequence you go, the larger its terms become, since Hn+1 = Hn + 1/(n + 1). However, they grow slowly, as the accompanying list shows: it gives the integer parts of several harmonic numbers. (In each case, n is the least integer for which Hn exceeds the indicated integer.) Thus, H4 is the first to exceed 2, H11 the first to exceed 3, H31 the first to exceed 4… and H40,427,833,596 the first to exceed 25!
| n | integer part of *Hn* |
|---|---|
| 4 | 2 |
| 11 | 3 |
| 31 | 4 |
| 83 | 5 |
| 227 | 6 |
| 616 | 7 |
| 1,674 | 8 |
| 4,550 | 9 |
| 12,367 | 10 |
| 1,835,421 | 15 |
| 272,400,600 | 20 |
| 40,427,833,596 | 25 |
All the way to infinity
Even though the harmonic numbers are in no hurry, they still tend to infinity. The first proof of this result, probably the best known, dates from the 14th century (see box).
