One plus a half plus a third, forever, never stops growing
Add one, then a half, then a third, a quarter, and so on through every unit fraction. The pieces keep shrinking toward nothing, so surely the total levels off. It does not. The harmonic series grows past any number you name, though so slowly that its climb tracks a logarithm.
The name comes from music. A vibrating string's harmonics have wavelengths equal to unit-fraction slices of its fundamental, and every term after the first is the harmonic mean of its two neighbours, another phrase borrowed from music. Baroque architects liked these ratios as well, using them to set the proportions of floor plans, facades and details in churches and palaces.
Nicole Oresme proved around 1350 that the series diverges, with an argument still taught today. Replace each fraction by a smaller one whose denominator is the next power of two. The new terms group into blocks that each total a half, and adding infinitely many halves goes on forever, so the larger original series must too. His work, alongside Richard Swineshead's on another series, marked the first appearance in mathematics of infinite series other than geometric ones, but it slipped into obscurity. Pietro Mengoli and Jacob Bernoulli published fresh proofs in the 17th century. The trick later grew into the Cauchy condensation test.
A second proof compares the sum with the area under the curve one over x. Stack rectangles one unit wide beneath the terms; the curve stays under their tops, and the area beneath it is an improper integral that never converges. That comparison also shows the partial sums, called harmonic numbers, stay within one unit of the integral, so they grow like a logarithm, offset by the Euler–Mascheroni constant.
Slow as it is, the series earns its keep. It appears in Euler's proof that there are infinitely many primes, in the coupon collector's problem about how many random draws complete a set, in the block-stacking puzzle of how far a pile can overhang a table edge, and in the average-case analysis of the quicksort algorithm.
Source: Harmonic series (mathematics)
