No, the sum of all the positive integers is not -1/12

I haven't been blogging recently because my new startup is taking up all my time, but someone needs to stand up and say the emperor has no clothes so it might as well be me.  About a week ago, two British mathematicians named Tony Padilla and Ed Copeland, who produce a video blog called Numberphile, posted a video that purports to prove that the sum of all the positive integers is -1/12.  It was making the usual geek rounds where I would have been content to let it circulate, but today the story was picked up by a naive and credulous reporter at Slate, where the story stands to do some real damage if not challenged.

(Aside for mathematicians: yes, I am aware that the claim is true under Ramanujan summation.  That is not the point.)

Let me start by recapping the argument.  Finding the flaw in the reasoning makes a nice puzzle:

Step 1: Let S1 = 1 - 1 + 1 - 1 ...

Then S1 + S1 = (1 - 1 + 1 - 1 ...) + (1 - 1 + 1 - 1 ...)

 = (1 - 1 + 1 - 1 ...) +
   (0 + 1 - 1 + 1 - 1 ...)

 = (1 + 0) + (1 - 1) + (1 - 1) ....

 = 1 + 0 + 0 ... = 1

So 2xS1 = 1.  So S1 must equal 1/2.

Step 2: Let S2 = 1 - 2 + 3 - 4 + 5 ...

So S2 + S2 = (1 - 2 + 3 - 4 + 5 ...) + (1 - 2 + 3 - 4 + 5 ...)

 = (1 - 2 + 3 - 4 + 5 ...) +
   (0 + 1 - 2 + 3 - 4 + 5 ...)

 = (1 + 0) + (1 - 2) + (3 - 2) + (3 - 4) + (5 - 4) + ...
 = 1 - 1 + 1 - 1 + 1...
 = S1 = 1/2

So S2=1/4.

Step 3: Let S = 1 + 2 + 3 + 4 + 5 ...

So S - S2 = (1 + 2 + 3 + 4 + 5 ...) - (1 - 2 + 3 - 4 + 5 ...)

 = (1 + 2 + 3 + 4 + 5 ...) -
   (1 - 2 + 3 - 4 + 5 ...)

 = (1 + 2 + 3 + 4 + 5 ...) +
  (-1 + 2 - 3 + 4 - 5 ...)

 =  0 + 4 + 0 + 8 + 0 + ...

 = 4 x (1 + 2 + 3 + 4 + 5 ...)

 = 4S

So S - S2 = 4S.  But S2 = 1/4.  So:

S - 1/4 = 4S

3S = -1/4

S = -1/12

Seems like an ironclad argument, doesn't it?  Like I said, finding the flaw in the reasoning (and there most assuredly is one) makes an interesting puzzle.  Here's a clue:

Let S3 = 1 + 1 + 1 + 1 ...

So S3 - S3 = (1 + 1 + 1 + 1 ...) - (1 + 1 + 1 + 1 ...)

 = (1 + 1 + 1 + 1 ...) - (0 + 1 + 1 + 1 ...)

 = (1 - 0) + (1 - 1) + (1 - 1) + ...

 = 1 + 0 + 0 + 0 ...

 = 1

But S3 - S3 must also equal 0, so we have just proven that 0=1.

The flaw in both cases is the same: the algebraic rules that apply to regular numbers do not apply to infinity.  Actually, it's more general than that: the algebraic rules that apply to regular numbers do not apply to non-converging infinite sums.  All of the sums above are non-converging infinite sums, so regular algebraic rules do not apply.  It is no different from using regular algebra when dividing by zero.  It doesn't work.

Now, there are ways to define the sums of non-converging infinite series so that they do not lead to contradictions.  The one that leads legitimately to the conclusion that 1 + 2 + 3 + 4 ... = -1/12 is called Ramanujan summation, which in turn is based on something called an analytic continuation.  But the problem is that the Numberphile video makes no mention of this.  They present the result as if it is legitimately derivable using high school algebra, and it isn't.  Telling people that it is does a grave disservice to the cause of numerical literacy.

UPDATE: Here's some more of the real math.