# I want to offer you an astounding result

@Cristi_Ichim (1821)

Bucharest, Romania

August 12, 2015 3:53am CST

If I were to ask you which is the result of 1 + 2 + 3 + 4 +...+ infinity (let's call this S) what would you tell me as an answer? You would probably tell me that it is obviously infinity, right? And that does make sense if we look at it in the terms that we are used to. But if we use a different perspective we will get something really different that seems very counter intuitive.
But in order to look at this from a different perspective, I would first need to show you how this series acts in relationship to other series. For example, I want to study how this series S acts in relationship to this series S1=1 - 2 + 3 - 4 + ...
So, what I am going to do is that I am going to subtract S1 from S and see what is the result of that. So:
S - S1 = 1 + 2 + 3 + 4 +...
-(1 - 2 + 3 - 4 +...) =
=0 + 4 + 0 + 8 + 0 +12 +... =4(1 + 2 + 3 + 4 +...)=4S
So, what I got here is that S - S1 = 4S. Well this is just a simple equation which implies that -S1=3S which implies that S=-(1/3)xS1. Well, yeah, but what is the result of S1?, you might ask...
Well, if I can determine the result for S1 then I can also determine the result of S.
Well let's have a look at S1!
S1= 1 - 2 + 3 - 4 + ...
If you stop at -8 then the answer will be -infinity but if you stop at +infinity then the answer will be +8. But does it stop at -infinity or at +infinity? I don't know. So, what I am going to do is that I am going to look at this series from a different perspective. I want to see which is the result of 4S1.
4S1 = 1 - 2 + 3 - 4 + ...
+ 1 - 2 + 3 - 4 + ...
+ 1 - 2 + 3 - 4 + ...
+ 1 - 2 + 3 - 4 +...
I don't know if it is visible but I put them in a way so that the second row is shifted one spot so that the number 1 falls underneath the number - 2 of the first raw. The third row is shifted the same as the second row and the fourth row is shifted two spots instead of just one. Why did I do so? Because this way this series can actually get a meaningful value. If you add the values on the columns you will get something like 1+0+0=1 and then all the other columns will go down to 0 because -2+1+1=0, -4+3+3-2=0, etc.
But this means that 4S1=1. But this means that S1=0.25. There are also other ways to prove that this series is 0.25.
But wait a minute if I know the result of S1 then I also know the result for S because S=-(1/3)xS1 which means that S=-(1/3)x0.25=-1/12.

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