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The sum of all positive numbers is....

Jayesh A Lalwani
Saloon Keeper

Joined: Jan 17, 2008
Posts: 2749


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....)
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


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


Why the RHS is same as S1


So S2=1/4

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


S-1/4= 4+8+12+16.....

the RHS is 4.S


So S=-1/12

Sum of all positive integers is a negative fraction. Is your mind blown or what?
Stephan van Hulst

Joined: Sep 20, 2010
Posts: 4642

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.

The mind is a strange and wonderful thing. I'm not sure that it will ever be able to figure itself out, everything else, maybe. From the atom to the universe, everything, except itself.
Paul Clapham

Joined: Oct 14, 2005
Posts: 19973

For the complete summary about why that sum looks valid but isn't quite kosher, see Bad Math from the Bad Astronomer.
Jesper de Jong
Java Cowboy
Saloon Keeper

Joined: Aug 16, 2005
Posts: 15100

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".

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Martin Vajsar

Joined: Aug 22, 2010
Posts: 3733

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".

Kaustubh G Sharma
Ranch Hand

Joined: May 13, 2010
Posts: 1283

I am maths blind :P

No Kaustubh No Fun, Know Kaustubh Know Fun..
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