Recapitulation

I’m back with another post after taking a break for a week. The last week of May was particularly busy as I was teaching a class called Introduction to the New Testament. During the week I also gave a lecture on Causes and Symptoms of Religious Discord, which you can find here if you’re interested. Anyway, back to Acutely Obtuse, we are in the middle of a four part series on the number e. In the last but one post, I started this series. We then saw why e is the base of the natural logarithm and the base of the exponential function. We also saw the relation between e and compound interest. In the previous post, we determined that e lies between 2 and 3. In the next post we will show that e is irrational. Given that e is irrational, it follows that it cannot be expressed as a ratio of two integers. This also means that it cannot be expressed as the sum of a finite number of rational numbers because the sum of rational numbers is necessarily rational.

Introduction to Infinite Series

Of course, this leaves open the possibility that e could be represented as the sum of an infinite series, that is a series that does not have a finite number of terms. In the previous post we looked at the limiting process and concluded that

Using sigma notation, we can write

This means that e can indeed be expressed as the sum of an infinite series. However, the above series is not the only one that has been shown to converge to the value of e. In this post, I wish to consider the above series and two others that have been derived for determining the value of e. However, while we just about managed to derive the first series without the use of any calculus, most of the series that have been derived require the use of calculus. Since I do not wish to introduce any advanced calculus at this stage in the blog, I will be considering one series that is derived from the first one itself and accepting, but not deriving, a second that has been derived using some advanced calculus.

But first we must ask ourselves why additional series are needed. If we already have a workable series, why should we bother to obtain more series? Mathematicians, after all, tend to be quite frugal, rarely doing more than is required. So why would they bother with more series representations for e when they already have one that one can use to obtain the value of e to any level of accuracy that might be needed?

Rationale for Other Infinite Series

To answer this question, we need to look at the values obtained by using the original series. Here are the first twenty values we obtain from the original series.

The red digits indicate where the current approximation for e begins to differ from the previous approximation. As can be seen, this series gives on average one additional digit of e for each additional term. Of course, for most purposes 10 or 12 digits of e is more than sufficient. In the table above, we have determined the value of e accurately to 18 digits. Why then would we need to determine more digits?

Computations of the digits of irrational numbers like π and e are used to measure the computational power of supercomputers. As we include more terms of an infinite series, two things must be done. First, the existing approximation of e must be kept in memory while the latest term is calculated. Second, the existing approximation and the latest term need to be added to obtain the new approximation. However, as we go down the terms in the series, each term becomes decreasingly small, requiring greater storage of memory. Since pushing things to and pulling things from memory are time intensive processes, at least from the perspective of a supercomputer, each new approximation tests the limits of the computational system to a greater extent.

However, once we have obtained the first (say) 18 digits of e, as we have above, it is pointless to use the same process to obtain the same results. Mathematicians hate repeating things because they know that they aren’t going to get new results. Hence, using the same series to obtain 1 billion digits from a set of computers to see which one is the fastest would be fine once. After that no new information is being generated.

However, mathematicians think, “If we are going to use infinite series to test the power of computational systems, why not generate series that give more digits per term?” In other words, if we can generate a new series that gives 2 additional digits per term, then with the same amount of time and same number of terms, we can generate 2 billion digits instead of 1 billion. And if someone can produce a series that gives 10 digits per term, then for the same price we can generate 10 billion digits. For people involved in the computational side of things, it does not matter which series is used. Any series can be used to arbitrate between competing computational systems. However, for mathematicians, the goal is to obtain new information. Hence, any series that promises more digits per term is to be valued for the additional information it can provide.

So mathematicians resort to all sorts of manipulations to produce new series that could provide more digits per term. The two that I have selected do precisely that.

Combining Terms

The approach for the second series recognizes that we can pair up terms and not affect the sum. Now, in general, consecutive terms can be written as

This term simplifies to 

However, since we have grouped the terms, n cannot now take all the values earlier specified or we will overcount. We can overcome this by replacing n with 2k to get

Hence, the original sum can be expressed as

Substituting values for k we get

We can see that each term is quite simple and that there is an easily recognizable pattern, which makes this easy to program. If we use this infinite series, we will get the following table:

Of course, as we should have expected, since we combined two terms of the original series to give a single term of the new series, the new series approaches the value of e twice as quickly as the original series. This means that we get, on average, 2 additional digits of e for each term of the new series. While this is laudable, as mentioned earlier, once this new series has been used, it will fail to produce any new information after a single use. This means that mathematicians will look for newer series to put into play. We could, as the approach to the second series suggests, combine more terms. For example, if we combine three consecutive terms we get

Replacing n+2 with 3k we get

This series obviously will converge to the value of e three times as fast as the original series. However, we can see that this process might become quite unwieldy. Finding the LCM of multiple terms is not a difficult matter. However, expressing them in a compact form that is conducive to programming is not necessarily a given. Already, while combining only 3 terms, we actually have quite a bit of deft algebraic manipulations to undertake. And remember, once a series is used, it pretty much becomes obsolete! Hence, we must discover new ways with which to express e.

Using Advanced Calculus

The third series I wish to consider is derived using the lower and upper incomplete gamma functions and the Taylor series. While the mathematics involved is certainly above the level of the blog at this stage, the end result is

If we substitute the values of k, we will get

Once again, each term is quite simple and we can easily recognize the pattern. So this too would be a good series to use for programming. But what kind of approximations of e does this series yield? The table below is instructive.

What we can see is that the third series converges slightly quicker than the original series but considerably slower than the second series we considered. 

Rejecting the Snake Oil

Over the years, mathematicians have derived many infinite series for e using all sorts of techniques. They have also used continued fractions, infinite products, and recursive functions. All these techniques are needed precisely because e is an irrational number, which we will discuss in the next post. However, as we have seen, there is little real world advantage to be gained from deriving more expressions for e. The requirement for new series comes from the realm of computation, where the computing power of a computational engine can be measured by having it perform processor intensive calculations such as are needed by these expressions for e. Mathematicians use this requirement to get the computational engines to churn out more digits of famous irrational numbers like e, π, and φ.

However, what we have seen in this post is something that we are rarely told about. In fact, I think what we have discovered is intentionally kept from us because it would reveal something we dare not admit. Now, hopefully, the mathematics involved in obtaining the original series, which was discussed in the previous post, was not too onerous. Granted that it was heavier than most other mathematical concepts I have dealt with so far in the blog, I believe it was not too difficult for most readers. The second series we considered was derived by combining the terms of the original series as pairs. Hence, this too did not require much advanced mathematics. However, the third series did require quite a bit of advanced mathematics and I avoided explaining it in this post. Despite this, the resultant series does not converge as quickly as the second series, which was obtained using much simpler mathematics.

What this tells us is that the level of mathematics involved in deriving an infinite series for e, or for that matter any irrational number, is no indicator of how quickly the series will converge to the desired value. This runs contrary to the intuition many people have about mathematics, namely that we need increasingly more complicated mathematics for more complicated problems. This is not the case and there are many instances when relatively simple mathematics can be put to use to solve extremely complicated problems. In my opinion, this is precisely what undergirds mathematical theorems like Gödel’s Incompleteness Theorems and some of the Millennium Prize Problems such as – P vs NP and the Riemann Hypothesis. It is also what lies behind the deceptively simple, but heretofore unsolved, Collatz Conjecture. 

In these days of increased dependence on Machine Learning, it pays to step back and understand that mathematics itself does not give us any guarantees or advice about what methods would work best for a given problem. Until we develop machines that are actually able to replicate human thought, any idea that more advanced computing capabilities would yield lasting solutions is, in my view, a pipedream. Unfortunately, too many of us are mathematically ill equipped to recognize when we are being sold snake oil in mathematical garb!