A strange derivation:
This sum 1+2+3+... = -1/12 is used in physics a lot believe it or not.
This can be rationalized sort of:
The Riemann zeta function is zeta(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + ...
where x^s means "x to the s'th power", and s is a complex number.
This series converges when the real part of s is >1. This function can be analytically continued over the whole complex plane, except for s=1, and this analytic continuation is called the zeta function.
Define a function yeta in TWO complex variables by
yeta(s,z) = 1/1^s + z/2^s + z^2/3^s + z^3/4^s + ...
The yeta function converges for |z|<1, for all s.
Then we have the functional equation
yeta(s,z)- 2z/2^s yeta (s,z^2) = yeta (s, -z), for |z| < 1.
When Re(s) > 1, we can take the limit as z -> 1, to get the equation
zeta(s)(1 - 2/2^s) = lim (z -> 1) yeta (s, -z)
Since the zeta function can be analytically continued to the complex plane except for a pole at s=1, the RH side can be analytically continued as well.
We have yeta (-1, -z) = 1 - 2z + 3z^2 - 4z^3 + 5z^4 - ... = 1/(1+z)^2
Taking the limit as z -> 1, we get
lim (z -> 1) yeta(-1,-z)=1/4.
So if the analytic continuation of lim (z -> 1) yeta (s, -z) = lim (z -> 1) yeta (-1, -z) for s=1, we have
(-3)zeta(-1) = 1/4, or zeta(-1) = -1/12.
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I thought this was neat but it didn't show up in the activity list.
Was the first line's sum supposed to be 1 - 2 + 3 - 4 + 5 - 6 + 7 -+ = -1/12?
No, it's all pluses.
1 - 2 + 3 - 4 + 5 - 6 + 7 - ... can be summed by Abel summation, one way of coming up with a sensible value for a divergent series.
Abel summation converts the sum into an analytic function
1 -2z + 3z^2 - 4z^3 ... taking the limit as z -> 1. This analytic function is the derivative of
-1 + z - z^2 + z^3 - ... = -1/(1+z), and the derivative is 1/(1+z)^2.
So by Abel summation, 1 - 2 + 3 - 4 + 5 - 6 + 7 - ... sums to 1/4.
By this same logic, zeta(0) = 1 + 1 + 1 + ... = -1/2 :)
Why would anyone ever use such an obviously ridiculous sum instead of saying, "Wait a minute, something must be wrong in the derivation of this sum"?
I don't know about 1 + 1 + 1 + ... but 1+ 2 + 3 + 4 + ... = -1/12 does get used in physics. I don't know exactly how this sum comes up, but it does. Rather than being infinity, it is -1/12. You don't want a value of infinity in physics.
The rationalization of these sums is that they are values of the zeta function.
There are many ways to come up with a value for divergent series that don't make sense in this way. For example, if you take the sum of a divergent series to be a limit of an analytic function as it approaches a point on the radius of convergence - this limit can depend on the direction in which that point is approached.
What I showed in the derivation above is what is "really going on" - expressing the manipulations done in the video, in mathematically correct form. I haven't yet proved the last step, though.
I'm sorry--I didn't realize there was a video I was supposed to watch. I see it now. I'll watch it. But although their method might very well yield -1/12, it isn't right. I say that with complete certainty. (For one thing, all of the partial sums are positive. For another, they're increasing. Fancy footwork can be impressive--but not when it gives an obviously false result. When such a thing happens, it's an indicator that something has gone wrong in the derivation. It's like deriving a contradiction in logic--you don't believe the contradiction; instead, you take the contradiction as a sign that something has gone wrong--false premises, invalid reasoning, *something*.)
Diagnosing the problem with their method will almost undoubtedly be beyond me. But I'll look at the video.
I would be *very* interested to know why physicists are using the result. I have a physicist friend--maybe he'll know.
OK, two things: First, we have to go back further than that particular series, because I don't buy that 1 - 1 + 1 - 1 + 1 - 1 + - ... equals anything at all. I have absolutely no doubt that some mathematical trick or tricks can be performed to make it appear that the sum is 1/2; but I don't think that that's going to get me to accept that the sum really is 1/2. I like mathematics--but it has to make sense.
Second, what was said just at the end of the video appalls me: the twenty-six-dimensionality of string theory comes out of that -1/12 sum? Then I don't trust the twenty-six-dimensionality (although I thought it was ten or eleven dimensions now)--not unless there's some derivation *not* depending on that sum. The fact that infinities arise in physics isn't grounds for thinking that that sum is finite. It's grounds for thinking that the physics isn't yet right (or hasn't yet been properly mathematically described).
I don't think that that's going to get me to accept that the sum really is 1/2. I like mathematics--but it has to make sense.
The video is a kind of mathematical tease.
Actually the sum 1 - 1 + 1 - ... IS 1/2 if you use Abel summation.
1/(1+z) = 1 - z + z^2 - z^3 + ..., and the limit as z -> 1 is 1/2.
It's grounds for thinking that the physics isn't yet right (or hasn't yet been properly mathematically described).
It makes the right predictions. Physicists often do wild and woolly things with math, and this is an example. What's going on doesn't seem to have been mathematically formalized. But you might guess there's an analytic continuation process implicit in the wild and woolly physicists' math, so that when they say 1+2+3+...= -1/12, what they really mean is that zeta(-1)=-1/12. If you look at the formula for the zeta function and write it down for s = -1, what you get is 1+2+3+...= -1/12.
Physicists often do things that make mathematicians unhappy.
I would buy this:
1 - 1/4 + 1/16 - 1/64 + 1/256 -+... = 1/ (1 + 1/4) = 4/5
1 - 1/2 + 1/4 - 1/8 + 1/16 -+... = 1/(1 + 1/2) = 2/3
and so on, so
lim (x->1) (1 - 1/x + 1/x^2 - 1/x^3 + 1/x^4 -+...) = 1/2.
However, I don't buy that the limit can be removed and the number 1 filled in for x.
Possibly more precisely, the first sum is really the limit of partial sums; the second sum is really the limit of partial sums; and the finishing limit--the one we're interested in--is really the limit of the limits of partial sums as x->1. While I agree that that limit of limits exists and is 1/2, I don't agree that the plain old limit exists when x=1. (In fact, I only agree that that limit of limits (1/2) exists from below.)
I don't agree that the plain old limit exists when x=1.
It's a matter of how you define the limit. One can define the limit of a sum by Abel summation. The limit by Abel summation exists for some infinite sums when the usual limit doesn't.
When the limit for an infinite sum exists in the usual sense, it has the same value as the limit by Abel summation, by Abel's theorem.
That's a crucial feature of these various methods of coming up with a finite limit for infinite sums. There are other ways besides Abel summation and analytic continuation.
The video isn't mathematically rigorous. I tried to show what is really going on in the video, to make it mathematically rigorous, in my initial post.
I'll happily grant that the Abel sum matches the usual sum (I assume you mean the sum considered as the limit of partial sums) whenever the usual sum exists. And I'll happily grant, on your say-so, that the Abel sum is -1/12. What I won't grant is that the Abel sum faithfully tells me what I'll get if I add infinitely many 1's together.
My position is somewhat analogous to that of someone who sees Alvin Plantinga's "Victorious" modal ontological argument for the existence of God, recognizes that it can't possibly demonstrate God's existence without seeing where its logic is flawed, and says, "It can't possibly work, because you just can't define God into existence." (I know what's wrong with it, so I can confirm that he's right.) I can't tell you what's wrong with Abel summation, but I can say that if it yields -1/12, then it can't possibly be right, because infinitely many 1's cannot possibly sum to something less than 1 (let alone something negative). -1/12 may well be the right Abel sum--but that can't possibly be the right sum.
I hope physicists are indeed using a harmless shorthand.
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