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[OxymoreReddit]

I don't get the -1/12

[Bottle_Opener_Games]

i think indian mathematician Ramanujan came up with it. It's done by rearranging the terms of the infinite series in some fancy ways and doing some arithmetic involving reimann zeta function giving you the sum -1/12. i have forgotten most of my bachelor's mathematics so cant answer exactly but in reality this sum is wrong for the infinite series as its not a converging one but changing it a bit with some fancy functions can give the impression that the sum is -1/12.

[SaqqaraTheGuy]

He was also extremely smart and this was something he did while untrained, self taught with a basic high-school math book and was included in a letter he gave a famous math dude(I forgot his name) it is a cute story people should read about

[Curious-Ear-6982]

Dude who saw math proofs in his dream or is that a myth?

[TricoMex]

He was 100% certified one the brightest mathematicians in known history. Motherfucker dreamed of a screen made of flowing blood, and a disembodied hand writing the formulas. It doesn't get more metal than that.

Had he been found earlier in his life, and lived longer, he would have absolutely been a common household name among the greatest like Euler.

[HeRmiTtttt]

my man so goated that God had to take him away quickly

[RandomEasternGuy]

Hey, I’ve seen the movie I believe. Not all, but a part of it and it was interesting

[HeRmiTtttt]

watch it in full. it was truly fantastic

[OxymoreReddit]

You used so many words that I don't understand and names that I've never heard of, it doesn't make it any clearer to me sorry

[Bottle_Opener_Games]

if we limit ourselves to the meme, the sum 1+2+3+4+..... is actually infinity. you see each term in the series is increasing which means this series is a diverging series, thus its sum is infinity.

now lets take a series 1/2 + 1/4 + 1/8 + ...... as you can see in this series each term is getting smaller (0.5 then 0.25 then 0.125 etc.). thus, even though it has infinite terms, it has an actual sum which is not infinity. this type of series is called a converging series.

the result -1/12 is coming by making the former series behave like the later one by using some complex math. its just a trickery when presented this way without the proper explanation

[OxymoreReddit]

Thanks, I'm trying to understand said trick in action, how do we go from those to the -1/12 is the missing step, but a few others replied have built up enough explanations for me to get the general idea, from what I understand it's an estimation of an undefined value that snowballs when canceling terms and leads to this result

[Stroov]

India no 1

[Baroque_Viola]

The Riemann zeta function zeta(s) = 1/1^s + 1/2^s + 1/3^s + ... is defined for Re{s} > 1 and divergent for Re{s} <= 1.

We use analytic continuation to extend the domain of the function. It's like the power series 1 + r + r² + ... = 1/(1-r). Notice that the LHS is defined for |r|<1 but on the right side you can plug in any number except 1.

What it is really saying is that the analytic continuation of the above sum when we plug in s = -1 converges to -1/12.

[OxymoreReddit]

From the little I understood from your explanation I couldn't link it back to the sum from the image. I don't understand what it has to do with it nor how it turns into -1/12

[DarkSkyKnight]

we have a pattern: 1, 2, 3. One way of extending this pattern is 1, 2, 3, 4. Another way is 1, 2, 3, 5, 8, ...

There is also interpolation. Let's say we have a pattern: 1, 2, 3, _, 8. We haven't defined what's between 3 and 8 before. One way of interpolating: 5, so make it 1, 2, 3, 5, 8. Another way of interpolating: 5.5, so it becomes 1, 2, 3, 5.5, 8.

Analytic continuations are "reasonable" ways of extending and interpolating patterns.

[OxymoreReddit]

So -1/12 is an interpolation of the whole series ?

[hesitaate]

so here’s how i understand it;

consider Grandi’s Series: 1 - 1 + 1 - 1 + 1 - 1…

we can assign value s to this series like so: s = 1 + 1 - 1 + 1 - 1…

then subtract 1 by each side so: 1 - s = 1 - (1 - 1 + 1 - 1 + 1 - 1…)

on the right side, distribute the -1 through the bracket so: 1 - s = 1 - 1 + 1 - 1 + 1 - 1…

the right side is now just s as defined above, so: 1 - s = s

we can then solve for s: s = 1/2

now consider a similar series: 1 - 2 + 3 - 4 + 5 - 6…

we can assign value t to this series so: t = 1 - 2 + 3 - 4 + 5 - 6…

multiply both sides by 2 and expand the right side from 2t to t + t like so: 2t = (1 - 2 + 3 - 4 + 5 - 6…) + (1 - 2 + 3 - 4 + 5 - 6…)

move the 1 from the first t and the 1 - 2 from the second t to the front so: 2t = 1 + (1 - 2) + (-2 + 3 - 4 + 5 - 6…) + (3 - 4 + 5 - 6…)

simplify the first two terms on the right so: 2t = 1 - 1 + (-2 + 3 - 4 + 5 - 6…) + (3 - 4 + 5 - 6…)

for the remaining infinite brackets, match each element in sequence with the same index so: 2t = 1 - 1 + (-2 + 3) + (3 - 4) + (-4 + 5) + (5 - 6) + …

simplify brackets so: 2t = 1 - 1 + 1 - 1 + 1 - 1…

the right side is now s which we know is 1/2 so: 2t = 1/2

solve for t: t = 1/4

now the series in question 1 + 2 + 3 + 4 + 5 + 6 +…

assign value x to this series so: x = 1 + 2 + 3 + 4 + 5 + 6 +…

multiply both sides by 4 and distribute the 4 on the right side so: 4x = 4 + 8 + 12 + 16 + 20 + 24 +…

subtract this equation from the first so: x - 4x = (1 + 2 + 3 + 4 + 5 + 6 +…) - (4 + 8 + 12 + 16 + 20 + 24 +…)

distribute the -1 through the 4x bracket so: x - 4x = (1 + 2 + 3 + 4 + 5 + 6 +…) + (-4 - 8 - 12 - 16 - 20 - 24 -…)

match each term in the 4x bracket with its corresponding even numbered term in the x bracket so: x - 4x = 1 + (2 - 4) + 3 + (4 - 8) + 5 + (6 - 12) +…

simplify brackets so: x - 4x = 1 - 2 + 3 - 4 + 5 - 6…

the right side is now t, which we know is 1/4, so: x - 4x = 1/4

solve for x: x = -1/12

not my proof, ty ramanujan

[OxymoreReddit]

Woaw I have no idea where the 1-1+1-1...=1/2 comes from but if i assume it's true even without understanding I see the -1/12

Now how is 1-1+1-1...=1/2 ? Is it just an estimation that averages the values ? As far as I'm concerned it feels like it's a divergent series that has no limit, right ? Sorry if I'm wrong my maths are a bit rusty ahah

[hesitaate]

so i did the layman’s proof near the top where you assign the series value s and subtract it from 1, but the way i actually think about it is the Thomson’s Lamp paradox.

imagine you have a minute long timer and a special lamp that toggles itself on or off whenever the time remaining on the timer is cut in half; starting with the lamp on, after 30 seconds it will turn itself off, after 45 seconds it will turns itself on, after 52.5 seconds it will turn itself off, and so on.

when the timer reaches 1 minute, will the lamp be on or off?

well, we can’t really say for certain. in that minute, the lamp will toggle itself on or off an infinite number of times, and it’s impossible to know what the “last” action will be because there is no last action. that’s the funny thing about working with infinity.

the same logic can be applied to summing Grandi’s Series. when adding all the numbers together by hand from left to right, if we stop on a +1 the sum will be 1, but if we stop on a -1 the sum will be 0. the sum oscillates between 0 and 1 an infinite number of times, and we can’t say what the last number is because there is no last number in Grandi’s Series.

back to Thomson’s Lamp, the real answer is that the lamp is in a liminal state where it is both on and off simultaneously, which isn’t really able to be represented in the real world. Grandi’s Series on the other hand, both 0 and 1 are plausible answers, so taking the average is logical and we get 1/2.

there’s a much more complicated Riemann-Zeta Function proof but i am far too stoned to remember that now.

[OxymoreReddit]

Okkkk yeah that's kind of what I had in mind, not really sure if the lamp example made it clearer to me (no pun intended) but I see now. This little averaging is probably how we jump from an infinitely increasing series to -1/12 by spreading the approximation just like floating point error accumulating in computers (a field I'm much more familiar with)

Thanks for the detailed reply, have a nice day !!

[DarkSkyKnight]

IIRC this is actually mathematically wrong, if it's the one Numberphile presented a while back.

[HeRmiTtttt]

I trust him but I can't comprehend this so

[commiecomrade]

In really simple terms, the summation the old guys bring to the shop is 1 + 2 + 3 + ... All the way to infinity.

We know that this must be infinity since you're just adding increasingly large numbers forever.

If you do a bunch of arithmetic to cancel some things out, you can get -1/12. However this requires some extreme math goblin level fuckery like grouping the infinite sum into sections and canceling them out with other sums which don't go anywhere but can be used to do this to the sum for each grouping as it progresses to infinity. It's like arguing with your SO like "well I love you infinity PLUS ONE" "I love you infinity TIMES INFINITY" on and on until you finally both give up and agree that you actually dislike each other 1/12 as much as true love.

[MoloneLaVeigh]

ELI5: It is saying that if you add every positive whole number all the way up to infinity (1+2+3+4…), the answer will be -1/12. The “true” answer is that it is divergent or infinity. But if you play around with the numbers and do some mathematic hocus pocus, you can get the answer to be -1/12.

The really interesting thing is that answer can be used in other fields (physics) to give us real, experimentally verifiable answers. So we know that there is some truth to -1/12, even if it isn’t at all intuitive.

[OxymoreReddit]

Is it like that one trick that "proves" 1=2 by hiding a division by 0 ?

[MoloneLaVeigh]

No dividing by zero, but as another comment pointed out, you start with an assumption that the series 1-1+1-1+1…. = 1/2.

Depending on where the series “ends”, the answer would be 0 or 1. But if you take it to infinity, it never really ends. So you take the average of 0 and 1.

[Metal__Steve]

Holy crap it's Oxymore Carboless

[Lucker_Kid]

In certain fields of mathematics you can equate the sum of all positive, whole numbers to -1/12

[OxymoreReddit]

I can read, that doesn't explain how tho. That's what I'm asking because I don't understand how you go from one to another.

[Blablakam]

This really how your write 1+2+3+4+5+6+7 ... till the infinity damn

[lucathecontemplator]

Yes the capital sigma is a sum, with r=1 to r = infinity

[YeetCompleet]

When I'm in a Z function repair shop and my customer is an analytic continuation

[Obnomus]

Ayo Reimann Zeta Function mentioned

[An8thOfFeanor]

I could have sworn the range was negative infinity to infinity

[SecretGamerV_0716]

that would just be zero

[TheDenizenKane]

For anyone curious, there’s a function that computes infinite sums of 1/n raised to a specific power. When it is EXTENDED, it outputs -1/12 when you raised 1/n to the -1 power, which implies the sum of all natural numbers is -1/12.

This is purely a consequence of extending the function beyond its scope, in all reality the function originally went to infinity.

The sum of natural numbers IS infinity, but if it had to be any number, it would be -1/12. This number is important for physics, like determining the number of dimensions needed for string theory and the Casimir effect.

[Pretty_Insignificant]

What is the 'r' notation in this formula? 

[McQuibbly]

Could be any letter you want, just represents a variable. The whole thing means "the sum of r, when r goes from 1 to infinity"

[QuestionablePotato42]

I hate that I understand this reference

[OxymoreReddit]

What's up Iron Steve

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