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The sum of all positive numbers is....

 
Rancher
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-1/12.

Apparently ramanujan proved it but couldn't believe the proof himself. He wrote to Hardy telling him that if he shows him the proof, Hardy would show him to the lunatic asylum.

The simplified proof is as follows ( at least how I understand it)

Let's say we have as sum S1 defined so


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

You could say
S1 = 1 - ( 1 - 1 + 1 - 1....)
So,
S1= 1 - S1
S1=1/2, which is kind of screwy to beging with.. But it's mathematically sound.. This is what happens when you start screwing around with infinite divergent series

Now let's take a different series

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

So let's say we add S2 to itself but we screw around with how the addition a bit

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

2.S2=1-1+1-1+1....
Why the RHS is same as S1

So,
2.S2=1/2

So S2=1/4

Now.. Let's take the sum of all positive number

S=1+2+3+4+5....

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

S-1/4=4.(1+2+3+4+5......)
the RHS is 4.S

So,
S-1/4=4.S

So S=-1/12

Sum of all positive integers is a negative fraction. Is your mind blown or what?
 
Saloon Keeper
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There are two problems though. 1/2 is only the Cesàro sum of the series. The real sum does not exist, exactly because the series is divergent. The second problem is that you may not simply rearrange terms when you add divergent series.

You only arrive at 2*S2 = 1-1+1-1+1 ... because you're lining up both series, and slightly offsetting one of them. This is valid for convergent series, but not for divergent. Using this method, I can also show that 1-1+1-1+1 ... sums to 6.
 
Marshal
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For the complete summary about why that sum looks valid but isn't quite kosher, see Bad Math from the Bad Astronomer.
 
Java Cowboy
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This all started recently with this Numberphile video:



And Phil Plain the Bad Astronomer wrote about it, and later posted a follow-up: Follow-up: The Infinite Series and the Mind-Blowing Result

And Scientific American also wrote about it: Does 1+2+3… Really Equal -1/12?

Conclusion: Calling -1/12 the "sum" of the series 1 + 2 + 3 + ... depends on what you mean by the word "sum".
 
Sheriff
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After reading a refutation of a refutation of an explanation of the original video, I decided not to weigh in with my opinion, except:

Jesper de Jong wrote:Conclusion: Calling -1/12 the "sum" of the series 1 + 2 + 3 + ... depends on what you mean by the word "sum".


Precisely!
 
Ranch Hand
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I am maths blind :P
 
My first bit of advice is that if you are going to be a mime, you shouldn't talk. Even the tiny ad is nodding:
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