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.

DragonLord Jan 30, 2020

#2

#3**+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.

Melody Jan 30, 2020

#5**0 **

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

Melody Jan 30, 2020

#6**+2 **

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

DragonLord Jan 30, 2020