Fun With Infinity

Infinity and infinite series and sets are concepts that stretch human intuition to the breaking point and as a result, are kinda fun - for masochists. The particular infinite series I'm gonna look at today is:

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... 

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.

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.

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 + ...

But that's the same as the 1st series that we already know an answer to:

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!