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# Just a fun question, not a serious one! Try it if you have some time.

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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

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never mind

Jan 30, 2020
edited by Guest  Jan 30, 2020
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The process repeats, I'm pretty sure all of them will be divisible by 9.

Jan 30, 2020
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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
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Oh, I see. Thanks for the response, Melody.

Jan 30, 2020
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Did you find this by yourself, or did you see it somewhere.  It is quite clever :)

Jan 30, 2020
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I was experimenting with numbers myself, to come apon this! It's quite interesting.

Jan 30, 2020
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Yes it is very interesting.  I'd love for someone to explain the error to us both.

Melody  Jan 30, 2020
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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
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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
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geno nailed it.

.

Jan 30, 2020
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Thank you for explaining me what the flaw is! Thank you!!!

Jan 30, 2020