1 + 2 + 3 + 4 + 5 + ... = -1/12

Sounds strange, doesn't it? I'll help prove it.

Let's declare a few series so that we can derive where this comes from. I won't bother to include all the reasons why this works (talking about convergence and whatnot), just so you get the gist. I'll do this first using summation notation, and then I'll show in plaintext what each of them means.

S1(z) = 1 + n=1∑∞{(−1)^n} [(n+1)^(−z) * (−n)^(−z)]
S2(z) = n=1∑∞{(−1)^(n+1) * n^(−z)}
S(z) = n=1∑∞{n^(−z)}

The simply declare some infinite series. I'll write them out below.

S1 = 1 - 1 + 1 - 1 + ...
S2 = 1 - 2 + 3 - 4 + ...
S = 1 + 2 + 3 + 4 + ...

Now that we have these, we can use them to help prove that S(z) is effectively equal to -1/12. The first thing I'm going to do is add the series S2(z) to itself (2S2(z)) so that I can equate it to one of the other series I've previously defined.

2S2 = (1 - 2 + 3 - 4 + ...) + (1 - 2 + 3 - 4 + ...) <-- this is shifted along slightly, i'll make that clear here:
2S2 = (1 - 2 + 3 - 4 + ...)
____+ (0 + 1 - 2 + 3 - 4 + ...)
2S2 = 1 - 1 + 1 - 1 + ...
2S2 = S1

Now we know that 2S2(z) = S1(z). So, let's solve for S1(z).

S1 = 1 - 1 + 1 - 1 + ...
1 - S1 = 1 - (1 - 1 + 1 - 1 + ...)
1 - S1 = 1 - 1 + 1 - 1 + ...
1 - S1 = S1
1 = 2S1
1/2 = S1

Now we know that S1(z) = 1/2, which means that S2(z) = 1/4. Here's where we get to see how 1 + 2 + 3 + 4 + ... infinitely must equal -1/12.

S = 1 + 2 + 3 + 4 + ...
S - S2 = (1 + 2 + 3 + 4 + ...)
____- (1 - 2 + 3 - 4 + ...)
S - S2 = 4 + 8 + 12 + 16 + ...
S - S2 = 4(1 + 2 + 3 + 4 + 5)
S - 1/4 = 4S
-1/4 = 3S
-(1/4)/3 = S
S = -1/12

Pretty cool, eh?

Keep in mind I obviously didn't come up with this myself, this is from a paper by Dr. Tony Padilla. Link here.

The idea of the first sum (S1) equaling 1/2, is hypothetical, it's the 'inbetween', the whole proof is based off of it. It's a nice idea, but mathematically, its only possible when assuming the answer to a sum that cannot be summed. Other than that, its amazing, used in string theory too.

The idea of the first sum (S1) equaling 1/2, is hypothetical, it's the 'inbetween', the whole proof is based off of it. It's a nice idea, but mathematically, its only possible when assuming the answer to a sum that cannot be summed. Other than that, its amazing, used in string theory too.

Of course this is extremely hypothetical, but there's not much else we can do with an infinite sum than to see how we can use, simplify and apply it (eg. String theory).

The idea of the first sum (S1) equaling 1/2, is hypothetical, it's the 'inbetween', the whole proof is based off of it. It's a nice idea, but mathematically, its only possible when assuming the answer to a sum that cannot be summed. Other than that, its amazing, used in string theory too.

Of course this is extremely hypothetical, but there's not much else we can do with an infinite sum than to see how we can use, simplify and apply it (eg. String theory).

It cannot be summed, so its use in string theory is beyond me. Still pretty interesting.

The idea of the first sum (S1) equaling 1/2, is hypothetical, it's the 'inbetween', the whole proof is based off of it. It's a nice idea, but mathematically, its only possible when assuming the answer to a sum that cannot be summed. Other than that, its amazing, used in string theory too.

Of course this is extremely hypothetical, but there's not much else we can do with an infinite sum than to see how we can use, simplify and apply it (eg. String theory).

It cannot be summed, so its use in string theory is beyond me. Still pretty interesting.

I have no clue why either. Still interesting though :P