Just saw above that you want to know the proof of S= 1+2+3+4+5..... = -1/12 (not positive 1/12 as stated above)
The reason this calculation is so important is that it attaches itself to probably the most important unsolved problem in mathematics for the past few hundred years which is the Riemann Hypothesis which is quite complicated and uses imaginary numbers to try and return the answer zero for the Riemann zeta function. (if you can prove/disprove this theory you win $1 million, just an FYI).
Anyway if you want to know the proof its probably a bit easier than you thought:
so the question is basically: ∞
SO this is stating that we are adding all the integer numbers up to infinity as shown at the top. To work this out were going to need a few things:
so first of all lets say that S1= 1-1+1-1+1-1+1-1+1............
The answer for this could be either 0 or 1 depending on where you stop it so you take an average of possible answers and find that S1=1/2
Secondly we say that S2= 1-2+3-4+5-6+7-8..........
To work out this we take 2S2 which is obviously 2 lots of what is stated above but we overlpa them to find a solution.
So we do 1-2+3-4+5-6+7)
As you can see if we add up the rows we get 1-1+1-1+1-1+1-1+1.... or S1 as we called it
So we know that 2S2= 1/2
and therefore S2= 1/4
So know we can work out S
if we do S-S2 we get something like this: 1+2+3+4+5+6.....
And if we take the rows as before we can see that you get 4+8+12+16......
As you can see if we divide that by 4 we get back to 1+2+3+4+5..... so therefore the answer to this is 4S
SO know we have the equation S-S2=4S but we know S2 is 1/4
So S-1/4=4S if we then take an S from both sides and then divide by 3 you S= 1+2+3+4+5....= -1/12
This theory is important to lots of mathematics and calculation in various things such as physics and economics. It is also the backbone of string theory which to be true needs 26 dimensions and you get this from 2 base dimensions of matar moving back and forth plus 24 extra dimensions (2*12). basically 12 pops up in a lot of advanced mathematics. This is only a basic proof of this and there are far more complicated proofs about if you want to read further.
Edited 3/24/2015 15:45:21