+0  
 
+2
159
11
avatar+278 

Have any of you guys wondered what \(\sum_{n=0}^{\infty}n\) is? (btw, this is the sum of all positive integers)

 

Most of you probably would start to think the answer might be \(\infty\) or \(\frac{-1}{12}\) (Ramanujan Summation).

 

I have a proof that its different. Tell me if there are any flaws.

 

Lets say that  \(1+2+3+4...=S\)

 

Then, we can add up three consecutive integers to get \(1+9+18+27+36...=S.\)

 

After that, we can factor our the 9's to get \(1+9(1+2+3+4...)=S.\)

 

This means we can say that \(1+9S=S\)

 

Solving the system of equation gets us that \(S=\frac{-1}{8}!!!\)

 

So the sum of all the positive integers is a negative number?!

 

Tell me if there is a flaw.

 Jan 30, 2020
 #1
avatar
0

 

never mind

 Jan 30, 2020
edited by Guest  Jan 30, 2020
 #2
avatar+278 
+2

The process repeats, I'm pretty sure all of them will be divisible by 9.

 Jan 30, 2020
 #3
avatar+110189 
+2

I do not claim to well understand this either but I think it does not work because

the sum of  9+18+27.....  is a bigger infinity than the sum of 1+2+3 ....

 

Scratch that, I do not know.  I have never accepted Ramanujan's summation, though as with yours, I could not see the logic fault.

I have not seen the fault in your logic yet either.

 Jan 30, 2020
edited by Melody  Jan 30, 2020
 #4
avatar+278 
+1

Oh, I see. Thanks for the response, Melody.

 Jan 30, 2020
 #5
avatar+110189 
0

Did you find this by yourself, or did you see it somewhere.  It is quite clever :)

 Jan 30, 2020
 #6
avatar+278 
+2

I was experimenting with numbers myself, to come apon this! It's quite interesting. laugh

 Jan 30, 2020
 #7
avatar+110189 
+1

Yes it is very interesting.  I'd love for someone to explain the error to us both. 

Melody  Jan 30, 2020
 #8
avatar+21955 
+1

I believe that the flaw occurs because you handle "S" as if it were a real number.

There is no real number (or whole) number that represents the sum of 1 + 2 + 3 + ...

 Jan 30, 2020
 #9
avatar+110189 
0

Thanks Geno

That is what I thought too.

You cannot say one infinity is equal to another infinity.

I still do not claim to understand properly  though.

Melody  Jan 30, 2020
 #10
avatar
0

 

geno nailed it.  

.

 Jan 30, 2020
 #11
avatar+278 
0

Thank you for explaining me what the flaw is! Thank you!!! laugh

 Jan 30, 2020

20 Online Users

avatar
avatar
avatar