Someone and Anyone

Currently, I’m reading John Derbyshire’s Prime Obsession. In this book, Derbyshire makes a very good effort to explain about the distribution of prime numbers and the Reimann Hypothesis in the layman’s language. Mathematics has been treated rather loosely at several places in the book, but you can forgive Derbyshire that. He has, in fact, tried to make mathematical concepts less mathematical and more intuitive in his book. The subject of the book is not central to the subject of this series of articles, though. In the next few articles, I am going to try and explain the concepts of convergence and divergence as simply as I can, using some examples from Derbyshire’s book. In this article, I’ll be talking about the concept of convergence.

Consider a finite series. For ex. 1 + 1/2 + 1/4 + 1/8. This series (essentially a sum) can be calculated precisely, because the number of terms in it is finite. The sum, in fact, is equal to 15/8 or 1.875. Any such finite series can be equated to a known quantity. However, when a series is infinite, i.e. it has infinitely many terms, precisely computing the sum is not possible — the series computed up to any large N can always be bettered by adding the (N+1)th term. In other words, it is not possible to equate the sum of an infinite series to a known quantity. So, the question is: can it be “approximated” at least? In other words, does it exhibit limiting behaviour towards some quantity? To put it in yet another way, does it tend toward some quantity? The answer is: Yes, sometimes. (At other times, you cannot zero in on a quantity for an infinite series at all. More on that in a later post.)

Consider the following infinite series now: 1 + 1/2 + 1/4 + 1/8 + 1/16 + …. Both, the finite series we saw earlier and this infinite series, have the same pattern of occurrence of terms (or progression). They are both geometric series, with 1 as the first term and 1/2 as the common ratio. Yet, these two series are entirely different in nature.

I know I have said that computing an infinite series is not possible. Even so, let us just start adding up the terms in the above infinite series. Let us see where it leads us. Up to four terms, the sum is 1.875, as we have seen earlier. The mathematical term of art for this is: the partial sum up to four terms is 1.875. Up to five terms, the sum is 1.9375. Up to six terms, 1.96875. Up to ten terms, 1.998046875. If you keep adding more and more terms like this, you will notice that the partial sum improves with the addition of more and more terms. However, you will also notice that the improvement in the Nth partial sum over the (N-1)th partial sum diminishes vanishingly as N increases.

Let us now take a “metrological” perspective of this — let us trace this infinite series on an imaginary six-inch scale/ruler. Let us assume that the Nth term in the series indicates the length (in inches, say) that we have to progress on the ruler. Assuming that we are at the zero mark to start off with, let us start moving along the ruler according to the value of each successive term. The first term is 1. So, on reading the first term, we progress to the 1-inch mark on the ruler. Since the second term is 1/2, we now move to the 1.5-inch mark. At the third term, we find ourselves at the 1.75-inch mark. And so on. Basically, at the Nth term, our progress on the ruler is half of that at the (N-1)th term. Hence, for infinitely large N, the progress on the ruler is infinitesimally small compared to the (N-1)th term.

As we can verify on the imaginary ruler, as more and more terms are added, the partial sum of the series gets closer and closer to the quantity 2 without ever equalling it. However, there is no limit to how close the partial sum of the series can get to 2. For any N, the Nth partial sum is closer to 2 than the (N-1)th partial sum. The larger N gets, the close the Nth partial sum gets to 2. For no value of N will the Nth partial sum be equal to 2, though. The mathematical term of art for this phenomenon is: the series asymptotically tends to 2. Loosely speaking, this means that the sum equals 2 at infinity. This is known as convergence. The series converges to 2.

We know that PageRank converges to the principal eigenvector of the modified adjacency matrix (L) of the Web. What this means is that the PageRank vector, no matter how many power iterations we conduct, will get painfully close to the principal eigenvector of L, but it will never be the principal eigenvector of L. However, there is no limit on how close the PageRank vector can get to the principal eigenvector of L.

There exist variants of convergence as well — absolute convergence and conditional convergence. (There are also pointwise convergence and uniform convergence, but I’m not yet fully equipped to explain them well.) But I think we’ll discuss those in another post, partly because I feel they might, by themselves, warrant a separate discussion and partly because my body is begging for some sleep right now.

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