Sum Numbers: More Than I know

By Bobby Neal Winters
I am teaching analysis once again out of A Radical Approach to Real Analysis by David Bressoud. It has done a lot over the years to broaden my understanding of the subject.  It has good examples, classical examples.  While many threads of the subject are explored, very good attention is paid to summing series.
We teach our students to do things without a second thought in introductory calculus that the ancient Greeks would’ve balked at, e.g. the summing of infinite series.  For example, consider the series 1+ 1/2 + 1/4 + 1/8+ ..., where the ellipsis indicates the terms go on forever. Note the pattern that each term is half of the previous term.  
This is an example of a geometric series.  Geometric series are distinguished by the fact that there are a common ratio between every term and the previous term. Such a series is determined by the first term and the common ratio.  The ratio in this case is 1/2. This particular example sums to 2.  Do the following thought experiment.  The number 1 of course is 1 less than 2; 1+1/2 is 1/2 less than 2; 1+1/2+1/4 is 1/4 less than 2.  The pattern emerges that, with each additional term, we are are splitting the difference between the current sum and 2.  In this way, any finite sum can be made arbitrarily close to 2.  We say that the series converges to 2.
But, as I said, this is just one example of a more general type of series.  We can consider the series 1+x+x2+x3+... .  Again, this is a geometric series because there is a common ratio between the current term and the previous is x.  This series sums to 1/(1-x).  there are careful ways to show this and less careful ones.  For the sake of ease, let s=1+x+x2+x3+... .  Note that we can write s=1+x+x2+x3+...= 1+(x+x2+x3+...)=1+x(1+x+x2+x3+...)=1+xs.  That is to say,  s=1+xs.  Solving for s in terms of x gives us, s=1/(1-x).  Therefore, 1/(1-x)=1+x+x2+x3+... .
If you are in a mood to do algebra, you can let x=1/2 in that formula, and observe the sum is 2, exactly as we argued in the first example.   I think that’s pretty cool, but then look at my life choices.
This is a keen little formula, but there are problems.  It won’t work for every choice of x.  If we let x=1, for example, then 1/(1-x) is one divided by zero, and as we have been taught from a very early age, dividing by zero is a no no.  It’s bad.  Don’t do it.  The 1+x+x2+x3+...  becomes 1+1+1+1+... which becomes larger with each additional summand and will, in fact, increase without bound.  We mathematicians say it “diverges to infinity.”
More of a problem is when we allow x=-1.  In this case, 1/(1-x) becomes 1/2, and 1+x+x2+x3+...  becomes 1-1+1-1+1-... .Here let s=1-1+1-1+1-... .  If we group the terms of the series  as s=(1-1)+(1-1)+(1-1)+...=0+0+0+...=0, so it appears we are done.  However, looks can be deceiving as we can also group them like s=1-(1-1)+(1-1)+(1-1)+...=1-0-0-0-...=1.  So we’ve proven that 1=0 which, though it might be useful at tax time, doesn’t really strike us at true.  
To muddy the waters even further, Gottfried Leibniz, made the observation that 1/2=1(1+1)=1-1/(1+1)=1-1+1/(1+1)=1-1+1-1+... and so forth, agreeing that the sum should be 1/2. This is halfway between 0 and 1, so in an age where we seem to seek compromise, 1/2 would represent meeting in the middle, making everybody happy.
In case you haven’t notice, though, mathematicians don’t put a high premium on making people happy.  
The lesson drawn from this is that one can’t simply deal with infinite sums the way one would deal with finite some.  Infinity makes a difference.  Mathematicians, again unconcerned about making people happy but deeply concerned about making sense, ferreted out conditions in which one could treat infinite sums more or less like finite ones.  They distilled the concept of convergent series.  The series 1+x+x2+x3+...  converges to 1/(1-x) only when |x|<1 .="." nbsp="nbsp" span="span">
For many, this closed the book on series that don’t converge, the so-called divergent series.  Yet such great lights as Leonhard Euler--which is pronounced Oiler not Yooler.  You sound like a hick when you say Yooler--used divergent series to get results which were, in fact, true.  This is insane. It is reminiscent of the joke about the woman whose husband thought he was a chicken.  She didn’t have him put away because they needed the eggs.
This is an area where my reach exceeds my grasp. Suffice it to say there are ways where this sort of thing can be made precise and consistent results can be obtained, but it comes at a price of having a different understanding of what the sum means.
The formula 1/(1-x)=1+x+x2+x3+...  gives and gives.  One can use it to prove that 1/(1-x)2 = 1+2x+3x2+4x3+... .  If we let x=-1, this yields 1/4=1-2+3-4+5-... .  The technical term for this sort of behavior is “nuts.” But it gets worse.  Let t=1-2+3-4+5-....  and let r=1+2+3+4+5+.... While one might harbor some sort of hope for the fate of t because the terms bounce back and forth between being positive and negative, r must be positive.  But wait.  r-1/4=r-t=2(2+4+6+...)=4(1+2+3+...)=4r.  By algebra r=-1/12.  So 1+2+3+4+...=-1/12.
There are ways of making this all precise, but it requires a different interpretation of the symbols involved than is usually given them.
And I’ve told you more than I know about this.

Making Those Distinctions: 2 is not 3  

By Bobby Neal Winters
As I’ve mentioned before, I am a low-dimensional topologist.  At least I was before I began spending my time shuffling papers.  Doe the quality of being a low-dimensional topologist persist over time?  Perhaps it can be proven or perhaps I am an example that it does not.  
But I digress.
My time away from doing mathematics has given me time to think about mathematics.  Does a eunuch think more about women than other men?  I am not sure I want to research this. In any case, when I was doing mathematical research, I didn’t take the time to reflect on it. Perhaps that’s why I wasn’t more successful.  Now that I am viewing it from a distance there are things I ponder.  One of the things I ponder is the methods by which mathematicians make distinctions.  
In low dimensional topology this comes to the forefront because so much of the subject deals with pictures and mathematicians distrust pictures.  Pictures can be deceiving either purposefully or accidentally.  As a result of this, we make a practice of translating things from pictures to something that is closer to words.  We create artificial languages in which we can carry on our arcane conversations.  Sometimes differences which are easy to see (but difficult to be sure of)  in the picture are difficult to see (but absolutely certain) in our symbolic language.
Consider the following two objects:

2-holed torus

3-holed torus

These are called the two-holed torus and the three-holed torus, respectively.  They are generalizations of the torus.  A torus is a surface that looks like a donut and I will omit the picture, assuming everyone has seen one.
These two surfaces are different.  I mean, look at them.  One has two holes and the other has three.  As an old professor of mine used to say, “A blind cow could tell them apart.”  Indeed, once it has been proven, we can say they are different, but such are the standards of the subject that we can’t just assume this a priori; it has to be proven.
Okay, how?
There are a number of ways.  Consider something called homology theory. This is complex (and that is a great pun for the in-group, but unless you want to spend a couple of semesters working at it, let it go). You begin by creating models of your surfaces using triangles.  One can then obtain abelian groups from these triangles.
Okay, I’ve just slipped in the concept of “group” along with the notion that there are special groups that are referred to as “abelian.”  Let it go.  Let it go, I say.
I can’t.  
A group is a mathematical object that has a binary operation that is modeled on multiplication.  The positive real numbers with multiplication are a group.  Those of you who have been abused by being taught about matrices will take comfort in the fact that two by two (n by n, actually) matrices with a nonzero determinant form a group.
Now, those of you who have studied two by two matrices may recall that they differ from real numbers because A times B is not necessarily B times A.   Groups where A times B is always equal to B times A are called abelian.  If you didn’t know that before, now you do.  Among those abelian groups there are free abelian groups.  To describe them would make this too technical and you might rightly fear that prospect.
One can use homology theory to associate a different abelian group to each of the surfaces shown above.  The two-holed torus can be associated to a free abelian group of order two and the three-holed torus can be associated to a free abelian group of order three.  
And, yes, the number of holes of the torus will always be the same as the order of the free abelian group that it is associated with.  It’s kind of happy (or ironic depending on your mood) that it turns out that way.  We are allowed to know that two does not equal three when we are in the world of free abelian groups, but we are not when we are just looking at the pictures.