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.