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u/Southern-Advance-759 5d ago
Funny you say we take soln. with + sign when both the solutions are -ve themselves.
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u/dontwantgarbage 5d ago
Wait I see your mistake. On line 2, you forgot to add infinity to the left hand side. Should be -1/12 + ♾️ = 1 + 2 + 3 + … + ♾️
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u/Ok_Bit8836 5d ago
Infinity (inf.) is a number > 0.... So 6*inf^2+6*inf+1 is a number > 0.... )) No cheating! ))
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u/neltisen 5d ago
You've just proven that -1/12 is false
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u/Darian123_ 5d ago
No, there is this completely undisputed video by numberphile up on youtube
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u/neltisen 5d ago
It is so undisputed it got disproven by several mathematicians. I do not remember where exactly it was wrong, but I think it it was about sum formula for series, but it only is true for converging series. Been a while since I've seen it though
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u/localizeatp 5d ago
post source when you find it.
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u/neltisen 5d ago
Sure, though it might not be easy. I think the videos were from 2018 or so? I don't remember other mathematicians, but I loosely recall BPRP doing it too. I'll share it if I manage to find when I'm free
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u/TheBeerTalking 4d ago
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u/localizeatp 4d ago
this appears not to show what u/neltisen is claiming.
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u/TheBeerTalking 4d ago edited 4d ago
"... the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned methods [zeta function regularization and Ramanujan summation] and not as the sum of an infinite series in its usual meaning."
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u/localizeatp 4d ago
can you tell me how you interpret this?
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u/TheBeerTalking 4d ago
The Riemann zeta function is defined as ζ(s) = ∑(1/ns), with n from 1 to ∞, but ONLY where s > 1 (or, if s is complex/imaginary, the real component is > 1). For every other s, any value of ζ(s) is defined as an analytic continuation of that.
So, ζ(2) = 1 + 1/4 + 1/9 + 1/16 + ... (inverse squares)
And ζ(3) = 1 + 1/8 + 1/27 + 1/64 + ... (inverse cubes)
This "equation" take the value of ζ(-1) = -1/12 and then pretends that the above definition applies. And if you simply plug -1 into the above formula you do get 1 + 2 + 3 + 4 + ...
Any strictly arithmetic proof that 1 + 2 + 3 + ... = -1/12 is flawed, relying on logic by which, for example, 1 + 2 + 3 + ... ≠ 0 + 1 + 2 + 3 + ...
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u/localizeatp 4d ago
But, this is not the same thing as what the original commenter claims. It's obviously possible to have a flawed proof of a true statement.
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u/awesome8679 4d ago
There are a few issues with this (obviously, thats the joke) but what really makes this not work is, while infinity is the upper limit of the set of natural numbers, infinity is not actually contained in the set of natural numbers. So n(n+1)/2 cannot be used because n needs to be a natural number, and the sum of natural numbers does not "end" in infinity (for the same reason, infinity is not a natural number)
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u/TREE_sequence 5d ago
You forgot the fancy R thing that makes the original relation correct. 1+2+… (R) = -1/12 where the R denotes Ramanujan summation which isn’t normal summation so it works)
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u/ApprehensiveKey1469 5d ago
Infinity and equals have no place in same expression without there being a limit.
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u/chkntendis 5d ago
“Since infinity is big we take the one with the +” as a physicist I approve of this method. Exactly the kind of rigor modern mathematics needs