S = 1 + 2 + 3 + 4 + ...

What is the value of S?

The NY Times recently had an article demonstrating that a possible answer is -1/12 (there's a more rigorous proof that shows the answer is indeed -1/12 but is beyond what I can show on a blog). I know that some of you studiously avoid the NY Times and therefore might not have seen it, so I'll duplicate it here with a little more explanation.

There's only one somewhat non-intuitive bit to the proof, so let me address that before I get started. The best illustration of this bit of non-intuition is called Hilbert's Paradox of the Grand Hotel:

Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms...

What this demonstrates is that if I have two infinite sets (such as rooms and guests) with a one-to-one correspondence between each pair of elements in the sets, I can shift over all of the elements of one of the infinite sets, leaving one element without a corresponding element in the other set (room 1 in the example above), yet have all of the other elements of both sets still have a one-to-one correspondence.Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from his current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room.

Okay, we need to find the values of some infinite series. The first one is:

S1 = 1 - 1 + 1 - 1 + 1 - 1 + ...

What is the value of S1? To find an answer, we add it to itself and do the hotel room operation above (in other words, shift over one copy of the series).

2 * S1 = S1 + S1 = 1 - 1 + 1 - 1 + 1 - 1 + ...

+ 1 - 1 + 1 - 1 + 1 - ...

---------------------------

= 1 - 0 + 0 - 0 + 0 - 0 + ...

or, 2 * S1 = 1

therefore, S1 = 1/2

The second series we need is:

S2 = 1 - 2 + 3 - 4 + 5 - 6 ...

And we start with the same infinite shift operation that we used on the previous series:

2 * S2 = S2 + S2 = 1 - 2 + 3 - 4 + 5 - 6 + ...

+ 1 - 2 + 3 - 4 + 5 - ...

---------------------------

= 1 - 1 + 1 - 1 + 1 - 1 + ...

2 * S2 = S1 = 1/2

therefore, S2 = 1/4

So now let's work on our original series. We'll subtract S2 to help us find an answer:

S - S2 = 1 + 2 + 3 + 4 + 5 + 6 + ...

-[1 - 2 + 3 - 4 + 5 - 6 + ... ]

---------------------------

= 0 + 4 + 0 + 8 + 0 +12 ..

= 4 * [ 1 + 2 + 3 + ... ]

The right hand side is now 4 * S so rewriting we have:

S - S2 = 4 * S

or (subtracting S from both sides)

- S2 = 3 * S

Since we know S2 = 1/4, we have

- 1/4 = 3 * S

or

S = -1/12

or

1 + 2 + 3 + 4 + ... = -1/12

This sort of proof, where the sum of an ever increasing series is a negative fraction, makes some people's heads explode. I hope you're not one of them. It's just a little fun with infinity!

## 6 comments:

Nice take, Bret.

Another little known (by the larger public) thing about this result is that it is not well defined.

I propose the following challenge: use the same reasoning (sums of undefined series) and find another value for S.

Clovis,

You did notice that I always wrote "an" answer as opposed to "the" answer. Yes, I'm aware it's not well defined.

Your challenge is easy-peasy. Instead of shifting by one element when calculating S1, shift by two elements. Then S1, S2, and S are all zero. Though still a non-intuitive result...

Bret,

I sure thought you were aware, the challenge was for the other readers. And you just ruined it :-)

I do not like much the way the NYT article (following countless similar ones out there) tries to make a greater mystery of how this calculation ends up showing up in string theory.

Infinite series are important in Physics way before string theory was a thing. And more importantly, there are particular reasons - physical ones, not mathematical ones - for why, out of the many possible results an undefined series may give, this particular one is the "right" one in Physics (or in the string theory calculations where it appears).

I am not complaining for petty reasons. A few good sophomore students knock on my door, now and then, utterly confused about this.

Clovis wrote: "... tries to make a greater mystery of how this calculation ends up showing up in string theory."

Half the reason I posted this was because of it's relationship to string theory. I knew it would get your attention! :-)

This sort of proof...makes some people's heads explodeThanks for this, Bret. I can't really follow it all, but you've made me nostalgic for simpler, gentler times when people's heads exploded over brainteasers rather than the nightly news.

Funny you mention brain teasers Peter. I received this one from my daughter as a Chinese New Year Gift.

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