There’s been a bit of a rumpus recently about this video from *Numberphile* which purports to show that

It got posted on Slate with an accompanying blogpost by Phil Plait, which then got shot to pieces by various incensed bloggers, and Phil now has a much better follow-up post.

I’m not going to rehearse the story again – the short version is that the equation above is a really bad and misleading way of communicating a really interesting and surprising result. But I thought I’d share another titbit which has a similar moral, namely:

Infinite series often behave in very weird ways.

This in turn has two corollaries, both amply illustrated in the story above:

Unexpected and cool things can happen.

but at the same time

It’s very easy to start talking nonsense if you’re notextremelycareful.

The video is about the series . This is what we call a *divergent* series, meaning that as you add up more and more terms, the series grows without limit. An example of a *convergent* series is 0.9+0.09+0.009+0.0009+… Here, as you add up more and more terms, you get ever closer to the fixed value 1. This is the *limit* of the series. Convergent series are generally friendlier beasts than divergent series, but the purpose of this post is to illustrate that even convergent series can do shocking things.

Humans are not used to infinite series. But we are used to adding up finite collections of numbers. If we have three numbers (A,B,C) we know that the order in which we add them up does not matter: A+B+C=B+A+C=A+C+B etc.. This sort of thinking is so natural and automatic that it is all too easy to transfer it to the realm of infinite series, where they may not hold – not even (necessarily) for convergent series.

This post is about the following series S:

The first obvious question is: does S converge? And the answer is yes. It turns out to converge to a number near 0.69. (More precisely, but unexpectedly and coolly, the limit is^{[1]} the natural logarithm of 2 or ( ln 2 ). But you don’t have to know what that means to follow the rest of the post.)

Now, let’s reorder the elements of S. Notice that the numbers on the bottom alternate between being odd and even. Let’s mix things up so that they go ‘odd even even odd even even’ instead, but being careful to keep the sign of every element the same at it was before:

To reiterate, we have not added or removed any elements here, just reordered the elements of S. But if you look carefully at this new series, you’ll see that the positive terms are all followed by the negative of their halves – I’ll group these together in brackets to make it clearer:

Now let’s compute the bits in the brackets and give a name to the new series:

T:

But what’s this? If you compare a term of T to the term in the corresponding position in the original series S, you’ll see that it’s exactly half. And indeed, the limit of T is exactly half the limit of S – around 0.34 (more preciely, ).

So just by rearranging the terms of S we have completely changed its limit! That might seem surprising, but Riemann’s rearrangement theorem tells us that this is the tip of the iceberg. It is actually possible to rearrange the terms of S so that it converges to any number you care to name – or so that the series diverges!

So, be very careful when handling infinite series! Here endeth the lessen. But to continue with the mathematics a little: this happens because S is what is termed *conditionally convergent*, that means that if you replace all its negative terms with their positives:

H:

you get a series which doesn’t converge. H is called the harmonic series; that it diverges was one of the first truly surprising facts that mathematicians discovered about infinite series. *Absolutely convergent series* (ones where the corresponding entirely positive series converges, as 0.9 + 0.09 + 0.009 +… does) are much better behaved, and Riemann’s theorem doesn’t apply there.

**Finally, a shameless plug:** All this material is covered in my book Maths 1001.

[1] This is the case of the series expansion for which is given by the *Mercator series:*

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It seems that with such alternating “convergent series,” we can separate all the positive and negative terms into two divergent series leaving us with, you should pardon, infinity – infinity = ?, in short, anything we like. In a sense, this really returns us–past the gymnastics–to the Great Crisis introduced by Weierstrass in 1872.

Now combine the history of Hilbert’s “On The Infinite,” Godel’s Incompleteness Thm., and finally the birth of algorithmic information theory–and finally add General Relativity and Quantum Mechanics (intuited by Hilbert to an extent in “On the Infinite”), and voila–you can construct a convergent infinite series approximately modeling a finite interrelationship in physical reality.

That is, this math is the drawing of a finite algorithmic information target bullseye about the intersection of the previously shot arrow and the tree. However, if there is a law of physics–an empirical pattern in our sensory perception (what axioms really are, per Bertram Russell and Kurt Godel) to how arrows tend to hit a tree, such infinite series constructs can be a useful minimal shorthand to describe it.

When you take the limit of the rearranged series, you are essentially making a different series. You cop out by saying that at infinity, you include everything, but that really isn’t the case, or your limits would be the same.

Yeah, you can argue around me by defining your terms and qualifying your logic, but if you subscribe to the view that there are larger and smaller infinities, your conclusion is faulty. Your modified series excludes some terms ad infinitum, so of course its limit is different.