Someone and Anyone

My earlier plan was to write a bit about the various types of convergence in this article. However, I think we would do better to understand divergence first. So, here goes. In the previous post, we saw an infinite series that converged. In other words, it exhibited limiting behaviour. (This infinite series was a geometric series with 1 as the first term and 1/2 as the common ratio. As a matter of fact, any geometric series, with a first term a and a common ratio r, converges. That is, the sequence of its partial sums has a limit. As we saw, for the geometric series in the previous post, the limit of the sequence of its partial sums is 2.) But, what of those infinite series that do not exhibit limiting behaviour towards any quantity? Such series (i.e. ones that are not convergent) are said to be divergent. The partial sums of a divergent series go on increasing without limit. Loosely speaking, the sequence of partial sums of a divergent series tends to infinity.

From the example infinite geometric series in the previous post, we can make the following observation:

Observation 1: If a series converges, then the individual terms of the series must tend to zero.

In that series, the tenth term is 0.001953125, the twentieth term is 0.0000019073486328125, the fiftieth term is (approx.) 0.00000000000000177636, and so on. Notice how the Nth term is tending to zero as N increases. So, Observation 1 seems true enough. And therefore, we can conclude that if the individual terms in a series do not approach zero, then the series diverges. However, the converse of Observation 1 is not true. If the individual terms of a series tend to zero, the series does not necessarily converge — it may diverge. A very good example of this is the following harmonic series: 1 + 1/2 + 1/3 + 1/4 + 1/5 + ….

In this harmonic series, the tenth term is 0.1, the hundredth term is 0.01, the millionth term is 0.000001, and so on. It is clear that as N increases, the Nth term in this series tends to zero. However, this series is known to diverge. And there happens to be a rather old but elegant proof of its divergence by Nicole d’Oresme, a French scholar (c. 1323 – 1382).

d’Oresme observed that (1/3 + 1/4) is greater than 1/2. Similarly, (1/5 + 1/6 + 1/7 + 1/8 ) is greater than 1/2. So is (1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16). And so on. By first taking two terms, then four terms, then eight terms, then sixteen terms, and so on, it is possible to group the series into infinitely many “blocks”, where each block adds up to a value greater than 1/2. No matter how many such blocks we consider, it is always possible to come up with the next, well-defined block. That is, there is always a value x > 1/2 waiting to be added, no matter how many blocks we have already added up. Loosely speaking, the sum of the entire series must therefore be infinite. That is, the sum of the series increases without limit. The series diverges. Quod Erat Demonstrandum.

This elegant proof by d’Oresme seemed to have been lost on the world for several centuries. Pietro Mengoli proved this result all over again in 1647, using a different approach. Forty years later, Johann Bernoulli proved it with yet another approach. Shortly after, Jakob Bernoulli came up with yet another proof! Neither Mengoli nor the Bernoulli brothers seemed to have known about d’Oresme’s fourteenth century proof. John Derbyshire asserts that d’Oresme’s proof remains the most elegant of all the proofs for this result, and is the one given in textbooks today.

PS: Johann Bernoulli was the father of Daniel Bernoulli (of Bernoulli’s Principle fame). Jakob Bernoulli (of Bernoulli Trial and Bernoulli Numbers fame) was Johann Bernoulli’s elder brother. That is one super-cool family, eh?

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Other articles in this series: On Convergence, On Domain Stretching, Conditional Convergence and Absolute Convergence