Ramanujan: Making sense of 1+2+3+… = -1/12 and Co.

by birtanpublished on September 5, 2020

a lot of you will be familiar with this strange identity here one plus two plus three plus dot dot equal to minus 112 now when you think about it this looks totally insane because what we're doing here on the left side is we're adding

Larger and larger positive values infinitely many of them so if anything this left side should add up to plus infinity and not to negative 112 for most people who look at this for the first time I think that what is just

Nonsense well except it comes up in a very very famous letter in mathematics a letter was written by this guy down there Srinivasa Ramanujan in 1913 was addressed to G H Hardy in Cambridge in the UK one of the most famous

Mathematicians at the time now a lot of people have heard about this letter but have not seen it so I thought I'll show it to you okay so here's the letter it's a long letter and it's basically lots and lots and lots of theorems

Mathematical theorems it's an enumeration of lots of theorems that Ramanujan says he's found and the most remarkable thing about this is that she has no real formal mathematical training unlike Hardy and so a lot of

These things are pretty amazing and towards the end of a letter well this strange identity here comes up in a couple of similarly strange identities around it and actually this letter got an invitation to to England

To work with Hardy so you know maybe we should have a really really close look at this and let's actually what I want to do today now I'm not the first one to do this numberphile has done a couple of

Videos on this and other people but I think I'd like to really try to do this properly so it's going to be a very long video and so to make this accessible to as many people as possible I've split it into four parts so four levels of

Enlightenment and you can pretty much stop anywhere and you know get something out of it this way so let me know how far you get with this video since our Ramanujan was a devout Hindu I thought I'll ask the elephant God to help us out

Here to lead us through these four different levels of enlightenment when it comes to these strange sums okay here's level one just do it now Ramanujan actually doesn't tell how he derives this strange value in his letter

To Hari it's just you know it's just there it says that's what I get now we have his notebooks and there is a one of the pages from the notebook where he actually talks about this in the previous page he also talks about this

Is more like an afterthought to what he does on the other page but this one I can actually do in a video like this so I keep it completely elementary I can do it this one here so let's have a close look at what he does okay so he would

Start by saying see and he doesn't call it a sum he says it's a constant C is equal to this strange sum and then he does a couple of New Relic logical deductions and at the end of that he gets to minus 112 not much so we should

Be able to do this right okay so here we go so he says there's something there that something I call it C so C is equal to the sum now if that something is halfway it's reasonable should be able to manipulate this as usual so one of

The things I should be able to do is multiply it by four so he does that so four times C is equal to four times one is four I put the four under the one or I put it under the two whatever this is go with

It then four times two is eight again I skip one put the eight then I always skip one and put down all the multiples of four here in the second line and now the next step is to subtract the bottom from the top

you we get C minus 4 C is minus 3 C 1 minus 0 is 1 2 minus 4 is minus 2 3 minus 0 3 4 minus 8 is minus 4 and so on so we get this nice sum here 1 minus 2 plus 3

Minus etc ok now comes something surprising now Ramanujan somehow sees that this is equal to well 1 divided by 1 plus 1 squared y 1 plus 1 square that's bound for us so if that somehow makes sense

Well then we can say minus 3 C is equal to 1/4 just solve for C we get minus 1/12 ok now we still need that step here I mean that's not clear at all right that's not clear at all so how do we get that that actually has two ingredients

The first ingredient is the draw manager knows that the 1 minus 2 plus 3 minus etc sum is equal to the 1 minus 1 plus 1 minus et cetera sum squared ok second ingredient is this year that the 1 minus 1 plus 1 etc sum is equal to 1 divided

By 1 plus 1 okay so let's just try to make sense of this first one so here we've got an infinite sum that's multiplied by itself how do we do this well you have to multiply every term at the top but every term at the bottom and

Then add up everything so let's just arrange it like this so this is the first sum here that's the second sum here now we do first term times first term that's 1 times 1 is 1 now first term times second term is

Minus 1 first term times third term is 1 and so on and that actually gives us this nice checkerboard pattern of plus and minus ones how do you sum up this checkerboard pattern here well you do it in diagonals so here we go we're gonna

Put in 2 diagonals and now what's the sum that we're actually forming here well blue diagonal is one green diagonal it's minus 2 yellow diagonal is plus three next diagonal is minus four and so on so this then clarifies what what he

Means by this what about the second step that the 1 minus 1 plus 1 etc sum is equal to 1 divided by 1 plus 1 well that comes from something that a lot of you will be familiar with this identity here some of the geometric series now this is

Not valid for all ah it's only valid if the RS kind of small right so it's in the range from minus 1 to plus 1 this actually works so for example if we choose R is equal to 1/2 then this whole thing turns into 1 plus 1/2 plus 1/4

Plus 1/8 and so on is equal to 2 so you know a lot of people would have seen this a lot of people will be familiar with this and I said this works in the range from minus 1 to 1 it doesn't work 4 minus 1 it doesn't work for 1 at least

Not in standard calculus but what Ramanujan now does and a couple of people before him actually is substitute R equal minus 1 anyway and when you do that what you get well here you get minus 1 here you get 1 here you get

Minus 1 and so on so that gives you this identity and of course if you substitute this into that you get exactly what Ramanujan wants ok now if you know any calculus you think like every single step here is totally insane so there's

These sums here aren't this one here that one here that one here with subtracting infinity from infinity so nothing he makes sense so I've got my Insano meter here at the left so let's give this a score of 4 on the in

Centimeter well insane yes but remember this guy has a genius and this guy here GH re well he's he's not a dummy either so somehow they kind of take these things seriously so we better have a close look

Okay so what does Ramanujan do here well he actually does something that's their love mathematicians do when they don't know what to do so often you know you're encountering expressions like this so somebody just kind of scribble down on a

Piece of paper something with lots of dots in it and to start with that's just a couple of symbols on a piece of paper and you don't really know what what they mean all right I don't really know what they mean

You know if there's just finitely many without the dots usually know what it means but if there's the dots in there you know it gets tricky so don't know what this means right now you can try and kind of just

Straightaway develop some theory that make sense of these sorts of things but often it's a better idea to just kind of do it okay so what you do is you don't know what what these things are so you just let them be some sort of unknown

Value and then you just kind of go for it so if there are anything reasonable you should be doing these sort of manipulations that maybe are not too wild you know and then maybe it gets you some ways

And now let me just demonstrate this this is kind of a very famous one so you've got this infinite sum this was actually a geometric series that we just looked at and what can I do here well if that's anything reasonable I

Should be able to multiply it just like what Ramanujan dust at the beginning here it multiplies before I multiply by ah okay so I multiply our times C then on the other side we get one time so I put R

Here then R times R squared I put it there and so on and that will give me this and then I drew exactly what Ramanujan does and subtract the bottom from the top so C minus R see there we go

Then here well here we're really lucky a lot luckier than the nura managin was his son everything here pretty much cancels out and the only thing that's left over is the one and then we can solve for C we

Get this okay and so what we've found here is that assuming that this somehow makes sense and that we can manipulate it as usual necessarily get that this has to be equal to 1 over 1 minus R and now later on in calculus when you

Actually really go for it you actually say well this does make sense when R is in this range but only then yeah so for other values for R like minus 1 you know that one here that doesn't make sense that gets discarded that's evil we don't

Want to know this we don't touch this in standard calculus but actually it turns out that you know if you if you just set up mathematics in a slightly different way this might actually make total sense so on a different planet that might be

Totally ok I could also try I equal to 2 for example in here and then you get something really strange which is very close to the sum that actually started with so it's 1 plus 2 plus 2 squared plus 2 cubed that's all positive values

Getting bigger and bigger should add up to infinity but this guy here says it's actually minus 1 so how does that work well turns out if you just on the right sort of planet there is mathematics that makes perfect sense in which this is

Actually true okay and now that was level one so did you survive so far perfect now if you want to actually know how we make sense of all these things to

Stick around for level two okay so in level two we're actually going to focus not so much on what happens on other planets but what happens on earth okay so level two is about following the rules what are the rules for dealing

With these sorts of things in in standard calculus what makes this right and all of these wrong well short answer is that this guy here on the Left which is called an infinite series is convergent and has some two whereas all

Of those guys here are divergent okay let's have a close look now at some point in time somebody wrote down this infinite sum and at that point in time actually nobody knew what that really means well you know what 1 plus 1/2 is

You know what 1 plus 1/2 plus 1/4 is but what does it mean to add up infinitely many things I'm to start with there you know there's no it doesn't have any meaning all right but you can try to give it some meaning so we'll just start

Adding okay so we've got one note it down we add 1/2 to 1 we get 3 halves and then we add one force to get 7/4 you just keep going and going like this and we get a sequence of numbers down here and it goes on forever and when we have

A close look at this sequence of numbers the sequence of partial sums then we see well they get closer and closer to two and actually two is is greater than all of them and in fact two is the smallest number greater than all of these these

Values see all of these partial sums and if you look at this expression well if it's to mean anything I'm going to Billy the most natural thing to say here is that this should be equal to 2 and actually it's a definition it's a you

Know a definition that mathematicians have made it's our choice to say that you know this sum here is equal to 2 it's a definition it's a very natural one it's you know probably the most natural one

But it's still a choice our choice to do it this way and not another way in fact there's another human choice in there that our human choice is always kind of in the background whenever you deal with these sorts of things is that we're

Actually always by default talking about real numbers okay so we're talking about real numbers it's how we make sense of things there would be other possible choices for the default number system that we use and on

Different planets maybe they use something different for example the surreal numbers or the hyper real numbers but for everything that we do here on earth as a default it's the real numbers also important to keep in mind

Now why was our choice of definition so good well because to a large extent you can actually manipulate these these sums just like you would manipulate finite zones you know this you know sometimes not but for example if you've got

Convergent series here and the summit's a then you can actually multiply to the right side here by something for example five and you get something convergent again the series that we get down here is conversion again and it converges to

Has a sum five eight as expected so that's really good or if you've got two of those guys right so infinite series some a infinite series some be we can term wise add things and that gives us another infinite series that converges

And the sum is a plus B perfect or we can subtract the bottom from the top and that will give us the right thing a minus B perfect so what that means in particular is that if this guy here is a convergent sum then all of these

Manipulations that we did here are perfectly okay and we get the right result all right so we multiply we subtract that's it works again and we get the right result now in standard calculus courses you actually usually

Don't get to this point there's actually some bits about our definition that are not so great so for example if you've got something that works convergent convergent and you multiply them together you would expect that you

Always get like 80-something convergent with to some a trance be and often that is the case but sometimes it actually isn't so for example this guy and here that's convergent has a sum but when you

Actually do the square list this is sum by itself that doesn't converge and it's a bit of a problem but let's keep this in mind okay so this is Matt's on earth so let's just have a look at these some zero why don't they work well let's have

A look at the first one here partial sounds we need partial sums so first partial sum is 1 1 minus 1 is 0 plus 1 is 1 minus 1 is 0 and so on so 1 0 1 0 1 0 that doesn't converge to anything that doesn't have a limit so

It's you know it doesn't have a sum in a normal sense so that means it's divergent what about the second one 1 1 minus 2 is minus 1 plus 3 is 2 minus 4 is minus 2 and so on also doesn't settle down forget about it last one 1 3 6 10

Kind of explodes all these series here on the left side are divergent they don't have a sum definitely no 1/2 here definitely no 1 force here definitely not 1 minus 1 12 stare hmm but when you kind of look at it these sequences that

We get you the sequence of partial sums you know they're very different so in standard calculus you kind of just take everything that's not convergent and just say well throw it away so you don't really look at the

Difference that you get here there's a you know graduation there's actually different sorts of divergence here so different sorts of divergence and this one's kind of a tame one this is what is still kind of oscillating around a

Common center here somehow and it kind of explodes you know maybe you want to capture the differences between these different sorts of divergence so let's have a look at the first one here so it's a very very famous one actually

First thing and you see in calculus pretty much saying that you know this doesn't work you know for various reasons now partial sums 1 0 one there is supposed to be equal to 1/2 well how do you get 1/2 out of one and 0

Well somehow to the average all right so average of 1 and 0 is 1/2 and actually there is a very neat way of defining the sum of one of these series in a different way so what we do is we don't stop with the sequence of partial

Sums but now what we do is we generate a second sequence of averages so average of 1 is well 1 average of 1 and 0 is 1/2 average of 1 0 and 1 is 2/3 and then 1/2 again and it keeps on going like this so we have another sequence of numbers here

And actually when you look at that one here well every second one is a 1/2 and then the red ones here they also converge to 1/2 to have a limit of 1/2 so overall the sequence that we're generating here has a has a

Limit that is 1/2 so in terms of the second sequence what we get is convergence and and it makes sense to say that in some ways this series here does converge to 1/2 and there's actually a special special name for this

Sum is our convergent if that sort of thing happens right if you've got a series you first did a sequence of partial sums and then you do the sequence of averages if the sequence of averages converges to a number then this

Number is called the Cesaro sum of the series that we're looking at so the series is Cesaro convergent and actually this different sort of convergence this different sort of attaching a sum to a series makes a lot of sense and really

On a different planet there's a different sort of civilizations they may actually have chosen this as their default definition of some over-over series could well be this sort of thing actually has just as nice properties as

Everything else so for example if a series converges in the standard way then it will also be Cesaro convergent so that's really good but obviously what we've seen here it says there's certain series that this guy

Here can some that normal you know normal people can't count some also this sort of thing works alright so if you've got something that Cesaro sums to a then five times that is that one in terms of Cesaro and the adding business works the

Subtracting business works and so what that means again is that you go back to the very beginning if in your version of mathematics you're talking about Cesaro Suns whenever you're talking about something like this then this

Calculation here will actually give you the right answer and it that's right it gives us the answer one half for one minus one plus one and so on further in calculus actually learn about these things they are actually very

Useful so they do not only make sense but also very useful even for the stuff that you know we define as making sense so for example before I told you that when you've got something that has a sum a and there we've got another ones then

B so the standard way in the standard way then the product not necessarily will converge the product sometimes diverges but when you add the product the indices are away you will actually always get what you expect a times B so

Actually adding in this Cesaro way gets rid of one of those not so nice things about Europe a definition of of sons and there's other things that cesàro summation takes care of so there's something called Faiers theorem which is

About you all right so now have a look at our in centimeter well on Planet Cesaro if Ramanujan writes a letter he will

Actually only get an insanity score of three instead of an insanity square for that we had before okay now let's have a look at the next sum here on our list there was the one minus two plus three and so on

Some if we look at the partial sums we get one minus 1 2 and so on so that doesn't work now let's just do this is our thing so the averaging when we do the averaging we get these guys here and you see well that doesn't work right

That doesn't converse it we've got every second one is zero these guys here the red ones they converge to they have a limit of 1/2 okay so if you actually step back here step back and look at the sequence of partial sums you see like

Every second one is a zero and then eventually the red ones will be indistinguishable from one half okay well okay so there's one force that we had before what does have 0 and 1/2 to do with one

Force well it's it's average right its average so we can actually kind of repeat our game so we'll just do the averages of the averages now so we do you know put down 1 and then we do the average of those two and I do average of

Those two any average of those 4 it gives us another sequence of numbers and that will actually converge to one force I'm not going to write down the numbers now but it's going to work and we can

Actually now choose this higher order Cesaro thing go to mean our Sun right so on a different planet again you know so on a different planet maybe on the planet of the blue aliens people define sums of of series in terms of this well

First second third sequence of numbers that associate with any series could do that and in a planet like this well the in centimeter here would show just a reading of – all right so we're on a planet like this you know that gets

More and more reasonable what what Ramanujan does there and we can actually keep on going playing this game so we had like the first sequence that's us second sequence that was my green alien friend who prefers the sort of summation

Then the blue alien friend who prefers the next one down but and of course we can do more and more kind of averages or the next one would be the average of the average of the average another sequence and these methods get more and more

Powerful so you kind of go up and up and up and up and up and you can some more and more this series but what about this guy here will we ever be able to get to minus 112 well if you think about it no right so we've got positive numbers here

We do partial sums that's got to be positive numbers again averages of positive numbers again positive numbers or whatever you do here you'll never get anything that will get into the minuses all right for our super sum I have to

Tell you a little bit about functions okay so here we've got x squared graphed over the positives now we can extend this nice smooth function into the negatives in infinitely many ways and I've just drawn like three of them so

There's we can extend it as x squared but also in many many other ways in a smooth way across zero here okay just keep that in mind now so far we're talking about the real numbers and the real numbers are usually represented by

The real number line now I also have to talk about the complex numbers which is usually represented by the complex plane so here there's some complex numbers if you've never heard of complex numbers check out some of these videos here

Before you watch the rest now everything I've said about standard summing of series cesaro sums average of averages sums stays true if you're actually considering series of complex

Thus you can also have functions complex functions defined on various bits of the complex plane and some of them are actually super duper nice smooth they're called analytic functions okay so if you've got for example a function

Defined on the right part here of the complex plane everything to the right of the imaginary axis how many different ways are there to extend this to the left here to the left part of the complex plane now these analytic

Functions actually really really really nice a lot nicer than real valued functions so if actually this is a continuation exists then it's uniquely determined so it's something very very special and it's called analytic

Continuation of our analytic function so these things are super duper nice ok let's have a look at one of those things so here is an infinite series it depends on a variable that so that that can be any complex number right now we can

Check for which complex numbers does this thing converge say in a standard way in the standard way it converges everywhere here in the yellow so for example at 1 that equals 1 if we substitute here we get this series and

We've already seen this before in the Apple paradox video that sums to lock – all right now since you've watched this video you probably come up with an idea straight away what if we don't sum in the standard way but if we sum like the

Cesaro way and actually something really really nice happens you get part of the analytic continuation defined like this so if you sum Cesaro you get everything here to the right which is really nice among other things you can figure out

What's the value of the analytic continuation here at 0 and what you get well at 0 we have well that's everywhere 1 1 1 1 1 so we get 1 minus 1 and we know what this is our sum of that that's one half okay and then what about

Averages of averages well that gets us everything up to here so this is plug in minus 1 here minus 1 mmm well that's that sum and so that tells us that the analytic continuation here is actually equal to 1/4 and then we do averages of

Average of averages and we get this guy there and we keep on going like this and actually kind of nice homework assignment what do these question marks stand for a bit hard all right so these things actually useful you know for

Defining the analytic continuation of this very important dershlit function now we'll change all the minuses to pluses that's the Riemann zeta-function so the holy grail in mathematics the

Riemann hypothesis is all about this guy here so it's very important to know what it is now the series just was a standard summation defines an elliptic function here to the right again of the purple line so what does Cesaro give us what

Does averages of averages give us what do all the other things give us do they give us the analytic continuation of this thing well sadly not really so don't really get anywhere with this but there is an analytic continuation here

And let's just see what we get here formally if we substitute minus 1 so if we do minus 1 we actually get our Ramanujan son the super son okay okay so what are we thinking now well to start with when we define these generalized

Sums we took the first thing that came to mind here but what if there's different ways of generalizing the standard way of summing series other ones right a bit more sophisticated ones maybe those will allow us to calculate

The analytic continuation of this sum up there and they actually are the are more complicated ones for example Ramanujan invented one and then well since since they are such sums maybe it's possible to then calculate these sums using

Ramana turds manipulation and it's actually true analytic continuation at minus 1 is minus 1/12 you now here's something very very very

Important those first couple of summation methods that I talked about they conceivably could really be replacements of the standard way of summing things on some planet now these more complicated summation methods like

Romana Jones one when you look at it it's got integrals in it and it's got like Suns in it and special numbers they only work in special context they work for special sorts of series they're really really useful there but it could

Never ever replace standard summation if at any point in time you get on your calculus test what is one plus two plus three plus four etc and you answer minus one over twelve almost certainly you will get zero marks whenever you get

Something like this there's a standard way of interpreting is something like this and the standard way involves you know the partial sums do the partial sums converge or not over the reals that's what's being asked here nothing

Else if you give any other answer it will be wrong okay let's keep that in mind so the last thing I want to talk about is again the geometric series because most of you will be familiar with this and most of you have been told

You know to disregard anything outside you know small values of that now this one actually works for for complex numbers too and it defines a nice analytic function here in the unit circle if you actually now some using

Cesaro it will also give you things on the unit circle so it will actually give you the analytic continuation on the unit sir what else well if you do averages of averages actually doesn't get you anymore but there's an analytic

Continuation of course of this function that's defined here and this analytic continuation is actually this sum here to be calculated before so again I mean what comes out of there is not nonsense it defines something it defines values

Of the analytic continuation of the function that's defined by this guy here but that's what it is you

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