The following problem may rekindle interest in the harmonic series: An absent-minded spectator has lost his reserved seat number in a theater with N seats. He enters first and sits down anywhere at random. The other spectators arrive one by one and take their assigned seats. Whenever someone claims the seat he is occupying, he gives it up and moves to another vacant seat. On average, how many seat changes will he have to make before finding his own seat? We'll return to this at the end of the article…
An ingenious idea ===================
One of the first surprises awaiting young mathematicians is that an infinite sum of strictly positive numbers whose terms become "smaller and smaller" can diverge—that is, grow larger than any quantity we choose! This has been a source of amazement for centuries. Yet Nicole Oresme, a 14th-century theologian, proved that the harmonic series—the infinite sum formed from the sequence H of reciprocals of the positive integers (that is, 1 + 1/2 + 1/3 + 1/4 + …)—diverges. His proof of divergence appeared in 1360 in Questiones super geometriam Euclidis. In his proof, Oresme groups the terms so that each parenthesized group has a sum greater than 1/2. Let Hn denote the finite sum of the first n terms of the harmonic series (so Hn denotes a positive number), and consider H16 to illustrate his ingenious idea: H16 = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 \+ 1/15 + 1/16 so H16 = 1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + (1/9 + 1/10 + 1/11 + 1/12+ 1/13 + 1/14 \+ 1/15 + 1/16). Look closely: (1/3 + 1/4) > 1/4 + 1/4 = 1/2. (1/5 + 1/6 + 1/7 + 1/8) > 1/8 + 1/8 + 1/8 + 1/8 = 1/2. (1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16) > 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 = 1/2.
Every parenthesized sum, then, is greater than 1/2. It follows that H16 > 1 + 1/2 + (1/2) × 3, and hence H16 > 3. Similarly, if N denotes the integer 2n, then HN > 1 + (1/2) × n, so as n tends to infinity, the harmonic series diverges.
Be careful, however, when grouping the terms of a divergent series! Otherwise, we might conclude, for example, that the series S = 1 + 1 + 1 + 1 + 1 + …, whose partial sums, Sn = n, increase without bound, converges to –1 (!) because: S = (2 – 1) + (3 – 2) + (4 – 3) + … and hence S = –1 + (2 – 2) + (3 – 3) + … = – 1. The great mathematician Niels Henrik Abel even wrote in 1826 that "divergent series are an invention of the devil, and it would be shameful to use them in a proof".