Take as our starting point this function, defined on the positive whole numbers:
All it does is add together the fractions above, stopping when you hit the fraction specified by your particular choice of n. As you increase n and thus add more fractions to the sum, you'll end up with a plot of the function that looks like this:
As you keep adding more and more fractions the sum will get larger and larger, but the rate of growth will be very slow. I've stopped plotting at n = 100, where the last fraction is of course 1/10000. But even though the growth rate keeps slowing down, how do I know it won't keep growing without limit at a slow rate rather than close in on a particular value? The integral test for convergence, that's how. The integral of 1/x2 converges and therefore so will the sum we're looking at.
But what does it converge to if you let n grow toward infinity? The proof is more trouble than it's worth for a post like this, so we'll skip the hard work and present the result by magic. Behold!
Now there's sorcery for you. Pi has to do with circles and trigonometry. What's it doing floating around here with the sums of the reciprocals of numbers multiplied by themselves? And why pi squared? And why the 6? Well, I can't give you an easy answerin the Sunday Function format, but if you have some time to kill with you could do a lot worse than digging through some of the books in the link to the proof above. Math is amazing, and its awesome power in physics and the rest of the sciences is only a fraction of the beauty to be found in pure mathematics.
This particular example goes much deeper. You might want to find the sums of the reciprocals of other powers, for instance. It will turn out that the final answer for each limit is given by the Riemann zeta function. And the zeta function is intimately connected with the prime numbers and the prime number theorem which was itself the subject of a previous Sunday Function.
You reach uncharted territory very quickly. Here be dragons, but the dragons of math are more friendly than most...
Its been a long time since calculus class for me, but im pretty sure I have some notion of why!
this function has something to do with Sin or Cos (I think Sin). Basic trigonometry then explains why pi and 6 can be involved :)
NaiveIntellectual: you have the right idea. It's actually a trick involving infinite series of sines and cosines called Fourier Series. The same trick lets you sum up the reciprocals of all even powers. The fun REALLY starts when you're working on the odd powers. It was shown in 1978 that the sum of 1/n^3 over all n is irrational. Still open whether its transcendental (like pi^2/6 is) or not. You go up to fifth powers, and we're in virgin territory...
sunday is the lord's day. you should rest and not post.
I don't think God will mind too much, since it's material from His other book...
Along this line, a great place to laugh at string theory is this post arguing that
1 + 10 + 100 + 1000 + ... = -1/9.
Dear Carl #5,
mentally limited people may laugh while the more intelligent ones learn what analytical continuation is because your sum, and many more complicated "seemingly divergent" sums, are actually paramount in theoretical physics.
Matt, it is actually not hard to calculate zeta(2)=pi^2/6 in a blog format, even in the comment format.
Consider the space of periodic functions with period 1. Take the function f(x)=x-int(x)-1/2. It's piecewise linearly increasing and periodic, between -1/2 and +1/2.
The integral of f(x)^2 over the period (e.g. the (0,1) interval) can be calculated in two ways. Either realize that it is an integral of (x-1/2)^2 over (0,1) which is easily computed to be 1/12 (a cubic polynomial, difference taken between 1 and 0).
On the other hand, I can decompose the function f(x) into sines and cosines. Only sines will appear because it is an odd function. It is easy to see that
-f(x) = C[ sin(Y) + sin(2Y)/2 + sin(3Y)/3 + ...]
where Y = 2.pi.x, to make the right periodicity and where C is easy to be seen to be 1/pi.
How do I see it? The coefficients can be calculated by "inner product" which become integrals of sin(nY).linear, and this clearly contains the coefficient 1/n.
Then the "squared norm" of the function can also be computed by summing the squared coefficients in front of the sines times the squared norms of the sines - which are all 1/2.
So we have
1/12 = 1/pi^2 * 1/2 ( 1 + 1/2^2 + 1/3^2 + ...)
That obviously implies zeta(2)=pi^2/6, after 2 and 12 are partially cancelled.
Carl, I'm going to preface this by saying that I'm in no way convinced that String Theory is an accurate depiction of the universe (although if someone comes up with an experiment to check it, none but the String Theorists will be more excited that I). Anyway, the method being used in Lubos's post is standard in mathematics, and really shouldn't be scoffed at. It goes back to Euler, in fact, though he couldn't rigorously justify it. It's just a matter of taking a series, writing it in closed form on some domain, and then analytically continuing the closed form. Saying that 1+10+100+...=-1/9 is no more radical than looking at the value of the zeta function for arguments with real part less than or equal to one, which is a necessity in number theory (in fact, the Riemann Hypothesis requires this very trick, just to state).
Charles, I agree that in some instances, taking divergent series is perfectly good mathematics, especially for analytic continuation. However, in a physical problem, one must find reasons for justifying the analytic continuation. I linked Lubos' long and detailed article because I think it is the best argument for his point of view.
The traditional "regularization" method in QFT is to introduce an energy cut-off, that is, an energy beyond which one ignores the calculation. This was an honest method of removing infinities in that it was understood that the theory did not say what was going on at higher energies. Unlike string theory, these low energy "effective field theories" do not make ridiculous claims to be a theory of everything.
The argument has been going on for a long time and how people feel about it amounts to individual intuition. In favor of my view on the subject, do a google search for Feynman + "dippy process", or look for what Dirac had to say about cancelling infinities.
From a sociological point of view, the old school has largely been shouted down, but, on the other hand, string theory has lost connection with reality and now is reduced to inane anthropic arguments about the 10^500 vacua, etc.
In real life, one ends up with infinites when one makes excessively idealistic approximations to a real object. Yes one can calculate with analytic continuation, but one always knows that in reality, none of the real parameters are infinite.