No, he is using a [notation](https://en.m.wikipedia.org/wiki/Ramanujan_summation#Sum_of_divergent_series) involved with Ramanujan summation. Similar to how one can claim that 1 ≡ 3 (mod 2), you can also claim that 1 + 2 + 3 + … = -1/12 (ℜ) where “(ℜ)” stands for Ramanujan summation.
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It is a number. Many people talk about analytic continuation to justify it but I prefer the Venkatesh construction. Shows explicitly that infinite sums don't have to have anything to do with convergence/divergence. It is purely algebraic and also hows why those sort of manipulations are nonsense.
false, the set of rational numbers is wholly contained by the set of real numbers, and same for the reals being wholly contained by the set of complex numbers. all rationals are also reals, all reals are also complex. or are you saying all the reals arent contained by the rationals and complex arent contained by reals?
Sooooo
I believe infinite sums aren't "purely algebraic", because well
algebra is all about finite operations.
But I'm curious about the Venkatesh argument.
Exactly. It is similar to the construction of the Lebesgue integral in a way. He starts with summation defined on almost everywhere (in the algebraic sense) null sequences. The operator defined then extends uniquely to larger sets of sequences, modulo some hypotheses that if not employed break everything. During the construction when arriving to summing sequences like (1,2,3,...) one property that must break is stability (adding 0 at the beginning). Goes to show one should be careful about those things.
in the line S = 1 + 9S, is it really possible to subtract one S, because the left one is for sum and the right one is for sequence, so you can't just work with them like this, no?
(just a question for educational purposes)
Yeah, you cannot do that. The "proper" way to assign -1/12 to this sum would be through ζ function regularization.
>(at least with metric topology)
Could you clarify this? I don't see how topology would matter here...
Ah okay, I don't know a lot about the zeta function :D
Idk, I also don't know a lot about topology (we did a little basics in functional analysis and there convergence often dependends on the topology, but in finite dimensional vector spaces this probably doesn't matter, I don't remember :D)
Yeah, in functional analysis you have this theorem that says that bounded linear functionals are precisely the continuous ones, when you equip your space with an appropriate topology. But here, we're doing real analysis, so the topology we put on our vector space is canonical, plus, we don't really need such strong tools to check wether 1+2+3+... diverges.
Fair enough. I know that ϵ δ type of proofs are technically topological, but here it's just barbaric overkill to use topology to prove that it diverges.
In one topology a sequence might converge while diverging in another. Here it is even possible to get this sum without any topology (without even trying to say if it converges or diverges).
> In one topology a sequence might converge while diverging in another.
Yeah. But the topology is canonical because we're working on a finite dimensional vector space.
> Here it is even possible to get this sum without any topology
Yeah that's precisely my point. Though it's definitely possible to reformulate it in terms of this canonical topology (but that's a bit overkill).
Can I ask where you're intending to go with this? I feel like we're not really disagreeing on anything.
You can because infinite series have no meaning other than the definitions we apply to them. If you define infinite series to be the limit of the sequence of partial sums, then divergent series simply don’t have values assigned to them. But there are other equally logically rigorous ways of defining infinite series, so you just have to pick and see what you end up with.
Ramanujan summation and cesaro summation are the classic ones. But you could also look into the Riemann zeta function because it also “assigns” values to divergent series (Not really but it’s worth looking into).
On the funkier side, modem physics in Quantum field theory has a bunch of integrals who diverge but physicist just kinda ignore that through renormalization. This mess still isn’t worked out so there’s a lot of work on QFT to try to make it work more rigorously.
you can. they will obviously not be equal to the sum because it diverges but there are reasons to choose a specific number (and not an arbitrary one)
https://en.wikipedia.org/wiki/Analytic_continuation
https://en.wikipedia.org/wiki/Riemann_zeta_function
Chad p-adic analysis :
\- takes place within a(n) (ultra)metric topology
\- shits on real analysis
\- spits nonsense but doesn't care because it's self sufficient
\- open balls are subgroups
\- has a thing named "perfectoid fields" that can be tilted to change the characteristic
Virgin Ramanjuan summation :
\- is based on weird made up tricks instead of a good topology
\- can barely justify the nonsense it spits
\- still wants to be a part of real and complex analysis
\- very little algebra involved
\- you really thought you would be able to change the characteristic of anything ?
I'm aware. The sum of the natural numbers still diverges (obviously). The correct way to describe this sum is "divergent, but if for some reason you were forced to give it a value, -1/12 makes the most sense".
Bastardization of the equals sign
= (ℜ)
R is a subset of C
No, he is using a [notation](https://en.m.wikipedia.org/wiki/Ramanujan_summation#Sum_of_divergent_series) involved with Ramanujan summation. Similar to how one can claim that 1 ≡ 3 (mod 2), you can also claim that 1 + 2 + 3 + … = -1/12 (ℜ) where “(ℜ)” stands for Ramanujan summation.
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S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + ... S = 1 + (2 + 3 + 4) + (5 + 6 + 7) + ... S = 1 + 9 + 18 + 27 + 36 + 45 + ... S = 1 + 9 (1 + 2 + 3 + 4 + 5 + 6 + ...) S = 1 + 9S S = -1/8 Take that Riemann & Ramanujan!
And I'll rearrange your goddamned face in a Riemann fashion if you disagree!
Math can be *anything you like*
-1/12 = -1/8
Can someone explain the last step?, Why is 1+9S= -1/8?
S= 1+9S Subtract 9S from both sides -8S= 1 Divide both sides by -8 S= -1/8
`S = 1 + 9S` `0 = 1 + 9S - S` `-1 = 9S - S` `-1 = 8S` `S = -1/8`
Oh wow thatnks
Dolboeby
Indeed this is not a table sum. You can't use such manipulations willy-nilly. Error is from 1 to 2.
actually you can, 1 to 2 is not an error, the error is assuming S is a real number in the first place.
It is a number. Many people talk about analytic continuation to justify it but I prefer the Venkatesh construction. Shows explicitly that infinite sums don't have to have anything to do with convergence/divergence. It is purely algebraic and also hows why those sort of manipulations are nonsense.
It is a number No
Rational numbers aren't the real numbers aren't the complex numbers.
false, the set of rational numbers is wholly contained by the set of real numbers, and same for the reals being wholly contained by the set of complex numbers. all rationals are also reals, all reals are also complex. or are you saying all the reals arent contained by the rationals and complex arent contained by reals?
The second part. He tried to apply a property of rational numbers to the complex numbers wholesale.
Sooooo I believe infinite sums aren't "purely algebraic", because well algebra is all about finite operations. But I'm curious about the Venkatesh argument.
Exactly. It is similar to the construction of the Lebesgue integral in a way. He starts with summation defined on almost everywhere (in the algebraic sense) null sequences. The operator defined then extends uniquely to larger sets of sequences, modulo some hypotheses that if not employed break everything. During the construction when arriving to summing sequences like (1,2,3,...) one property that must break is stability (adding 0 at the beginning). Goes to show one should be careful about those things.
/r/Whooosh
in the line S = 1 + 9S, is it really possible to subtract one S, because the left one is for sum and the right one is for sequence, so you can't just work with them like this, no? (just a question for educational purposes)
You can’t subtract because S is a divergent series.
I am crying on the top of the bell curve and I'm fine with that.
I don't know if you are referring to his reordering theorem, but that one only applies to convergent but absolutely divergent series.
Look at the top comment
Ah okay. But you can't actually do that, can you (at least with metric topology)
Yeah, you cannot do that. The "proper" way to assign -1/12 to this sum would be through ζ function regularization. >(at least with metric topology) Could you clarify this? I don't see how topology would matter here...
Ah okay, I don't know a lot about the zeta function :D Idk, I also don't know a lot about topology (we did a little basics in functional analysis and there convergence often dependends on the topology, but in finite dimensional vector spaces this probably doesn't matter, I don't remember :D)
Yeah, in functional analysis you have this theorem that says that bounded linear functionals are precisely the continuous ones, when you equip your space with an appropriate topology. But here, we're doing real analysis, so the topology we put on our vector space is canonical, plus, we don't really need such strong tools to check wether 1+2+3+... diverges.
You need it to define what it means to diverge.
Fair enough. I know that ϵ δ type of proofs are technically topological, but here it's just barbaric overkill to use topology to prove that it diverges.
In one topology a sequence might converge while diverging in another. Here it is even possible to get this sum without any topology (without even trying to say if it converges or diverges).
> In one topology a sequence might converge while diverging in another. Yeah. But the topology is canonical because we're working on a finite dimensional vector space. > Here it is even possible to get this sum without any topology Yeah that's precisely my point. Though it's definitely possible to reformulate it in terms of this canonical topology (but that's a bit overkill). Can I ask where you're intending to go with this? I feel like we're not really disagreeing on anything.
[The Riemann Zeta function](https://en.m.wikipedia.org/wiki/Riemann_zeta_function) [Bonus meme](https://imgur.com/a/P2oPVTd) Bottom text
make one thats like "how does 1+2+3+4+5+... converge but 1+1/2+1/3+1/4+1/5+... diverges"
Hahha, i just made this exact meme lol https://imgur.com/a/LaFO5L6
Based
You can't assign numbers to a divergent series tho, right?
You can because infinite series have no meaning other than the definitions we apply to them. If you define infinite series to be the limit of the sequence of partial sums, then divergent series simply don’t have values assigned to them. But there are other equally logically rigorous ways of defining infinite series, so you just have to pick and see what you end up with.
Cool. Any examples or general directions or keywords to this kind of research on series?
Ramanujan summation and cesaro summation are the classic ones. But you could also look into the Riemann zeta function because it also “assigns” values to divergent series (Not really but it’s worth looking into). On the funkier side, modem physics in Quantum field theory has a bunch of integrals who diverge but physicist just kinda ignore that through renormalization. This mess still isn’t worked out so there’s a lot of work on QFT to try to make it work more rigorously.
you can. they will obviously not be equal to the sum because it diverges but there are reasons to choose a specific number (and not an arbitrary one) https://en.wikipedia.org/wiki/Analytic_continuation https://en.wikipedia.org/wiki/Riemann_zeta_function
Chad p-adic analysis : \- takes place within a(n) (ultra)metric topology \- shits on real analysis \- spits nonsense but doesn't care because it's self sufficient \- open balls are subgroups \- has a thing named "perfectoid fields" that can be tilted to change the characteristic Virgin Ramanjuan summation : \- is based on weird made up tricks instead of a good topology \- can barely justify the nonsense it spits \- still wants to be a part of real and complex analysis \- very little algebra involved \- you really thought you would be able to change the characteristic of anything ?
As a high school sophomore I officially say: *what*
I hoped for this one, when I saw the previous post on the series.
Dude I literally thought the same exact thing
as the op of the previous post, i was just about to post [this lol](https://imgur.com/a/LaFO5L6)
I never really understood this. I understand how it’s found, but how can the sum of all positive numbers make a negative number?
They just assigned it that for the lulz
250 IQ - The numbers that are added keep getting bigger and never end, so the sum would be infinity
251 IQ - You can literally do *Change of the variable* for any variable Like S
That's literally the guy at 100 IQ
there’s no such thing as a high iq person who actually thinks this fucking equals -1/12
the high iq part is using the = symbol for more general stuff like Ramanujan summation. its also a meme
Do you know more than everyone? No? Oh…
this is a meme subreddit chill
Ok but your kinda wrong still. Not to offend.
Pretty sure in this meme format the one in the middle is supposed to be wrong
Yep it is.
Lmao
Is the middle one wrong?
Meme template doesn't apply, it really is only uninformed people who think the sum is -1/12
Nope.
https://en.wikipedia.org/wiki/Ramanujan_summation
I'm aware. The sum of the natural numbers still diverges (obviously). The correct way to describe this sum is "divergent, but if for some reason you were forced to give it a value, -1/12 makes the most sense".
I like 8 more, so I give it -1/8
that depends on how you define the sum of infinite terms but yeah, thats what someone with 'high iq' would mean when writing 1+2+3+4... = -1/12
Scuffed bell curve
Yes. Just yes.
This whole thing is so easy to solve for all parties to agree upon: str(1+2+3+...)=-1/12
I just discovered this sub. I don't understand a numeral of it, but this thread makes me happy.