Riemann’s paradox: pi = infinity minus infinity

by birtanpublished on September 3, 2020

today I want to let you in on a really neat trick that I learned from a very famous mathematician how to subtract infinity from infinity to get exactly pi so for that I will return to this blackboard here that I found in The

Simpsons and I discussed in this video up there and particularly that that infinite series down there at the bottom yeah so it's 1 minus 1/2 plus 1/3 minus 1/4 plus 1/5 all the way to infinity in infinite series with some log too so

That's about 0.69 what I want to do here is something that somebody discovered about 150 years ago so you know just see what I do it's a bit of a paradox so what I do is I take a bit more of this and I really need to hold boil so we'll

Take out every fourth term and then what's left over has Otto nominator and even denominator terms that separate those are two now we're going to work our way in from the left take those two guys and put them down

There take those two guys put them down on the gap they chose two guys and put them down the next gap and you keep on going like this and obviously when you do this you use up all the terms that were there in the beginning and just

Rearrange them here and we bracket them a little bit now we're going to work our way through the brackets so first 1 here 1 minus 1/2 is 1/2 1/3 minus 1/6 is 1/6 and so on right so you can see how this works nice pattern here that's what what

Happens so we've just rearranged the terms and weari bracket at them a little bit and that should not change anything about the sum of this thing all right so if you compare what we've got now to what we started with well the bit that

We started with is what log 2 but now we can compare term by term so 1/2 of 1 is 1/2 1/2 of 1/2 24th and half of 1/3 1/6 and so on and so what you see is that term by term the bit at the bottom here is always one

Half of it the top so what it means is it should really be equal to one half log 2 now if you put all the stuff together you get blocked 2 is equal to 1 half log 2 which really amounts the same thing as saying that 1 is equal to 2 and

Whenever you come across something like this you start worrying there's really you know two different options here first one is Matt's it's broken or we made a mistake what do you think is more likely here well yes we made a mistake

And so where's the mistakes let's have a close look so there was three different identities so here here and there so two of these are correct and one of these is not correct so the first bit what I already said is and just believe me this

Is equal to log 2 then the bottom bit is also correct in fact that bracketing here and I just say this now you can put in brackets any way you want it's not going to change anything about this sum here is always going to add up to 1 half

Log 2 it's correct so what's really messing things up is that we are rearranging the terms so we're changing the order of the terms and that really creates a infinite series with a different sum than the one before

So basically when we're dealing with these infinite series unlike with the finite ones order matters rearranging the terms may change to some okay well some people are getting really worried oh no matt is broken anyway no it's not

Broken we just have to get used to this ain't gotta be careful about disorders so we just saw that we can rearrange the series into another series there's a different song and actually there's a famous mathematician riemann bernhard

Riemann who came up with a theorem the riemann rearrangement theorem which says that well in this case of this particular series you can rearrange this thing into series that add up to anything you want

For example pi and i'm going to show you how that is done how he how he makes up these different sums now you've probably heard this name the Riemann hypothesis is at the moment the most sought-after thing that you can prove in mathematics

Eternal Fame awaits if you can do this and that's all about the Riemann zeta-function and there's also the Riemann integral which you use all the time and lots and lots of riemann named bits in

Mathematics so this guy's is a mathematical superhero usually when it comes to these things that are named after him have quite difficult to explain properly but that riemann rearrangement theorem and him to do like

This with my animation magic so let's just do it but before we do it I really have to remind you of what it actually means for one of those series to add up to a certain number okay so what we do is we translate this series into a

Sequence of numbers the partial sums the sequence of partial sums so the first partial sum is just the first term just one the second partial sum is just those two guys added up that's point five in this case then third partial sum fourth

Partial sum and so you get the sequence of numbers here and in the case of this series the sequence of numbers converges to a certain number which is block two and then we say that the series has some that number log two in this case so

That's the definition of a sum of an infinite series so this is an example of a convergent series a series that actually has a sum which is a finite number now there's also lots of series that

Don't converge and they're called divergent so for example if you replace all the minuses in here pluses are all shortened that other video there that this adds up to plus infinity but there's other types of divergence so for

Example if we take this guy there it's 1 minus 1 plus 1 minus 1 the partial sums here are 1 0 1 0 1 0 they don't settle down to anything not finite not infinite that's also divergent so different sort of diversion to explain what's going on

Here let's have a look at the positive terms and negative terms separately so if you just add up all the positive terms you'll actually find you get infinity and if you actually add up the negative

Terms you get minus infinity and actually if you've watched the other video should be able to prove that yourself and maybe do it in the comments so what we're really doing here is we're subtracting infinity from infinity in a

Controlled manner what remands rearrangement theorem now says is that if you've got a convergent series who's positive terms add up to infinity and who is negative terms add up to minus infinity then you can

Rearrange the series into series that have any sum you want so let me demonstrate this now for pie with this particular series okay so we want pie that's positive so we're going to start with some positive terms so the first

Term of my new serie is going to be 1 so the first partial sum is also going to be 1 so partial sums I put up there well that's not enough so we add a few more of those positive terms let's get close to 2pi now we actually

Have to add quite a few of them to get close to PI we're still just under here so let's add one more that gets us just over this string of terms we choose as the beginning bit of our new series so we're just over now

We're going to use some negative terms to get just under okay so we use the actually the minus 1/2 is good enough so that gets us under 2.6 and so on alright now let's go over again so for that we'll just do some positive terms

Actually need quite a few again but we can't be absolutely sure that we can get over because at any stage of this process the positive terms that are left over will add up to infinity and the negative terms that are left over will

Add up to infinity so I can always be sure that no matter how far under or how far over I can always take enough terms to get over all right now we could just keep on going like this so then a negative term to get us under again then

A positive positive terms again to get us over and go flip back and forth and eventually we get to PI and we can also be sure that we get to PI because the terms themselves they get smaller and smaller and magnitude so

That means that as I overshoot and undershoot I get closer and close to PI and actually I can rearrange this initial series in infinitely many ways different ways to get me pi I can also rearrange it to get me an

Infinite sum all sorts of other things so maybe you also want to do this in the in the comments figure out the details there now for other series you can actually have the situation where nothing changes no matter what you do so

There's lots of convergent series where the positive terms add up to a positive number and the negative terms add up to a negative number so for example could be you know to the positive terms and minus seven the negative terms and then

No matter how you reshuffle this series the sum will always play to minus 7 minus 5 ok now there's something else that we're actually doing here we're actually bracketing and so we're betting so we can bracket like there so we can

Break it like that or maybe we don't do any brackets whatsoever and actually as long as we're dealing with infinite series that converge either this type of that type doesn't matter what you can put in brackets any way you want and

Some won't change so I mean that's that's a bit reassuring so basic comet activity is broken but associativity at least for convergent series doesn't get mucked up that's quite nice but there's there's at least one more surprise

Happening here that's just for convergent series so if you actually look at some divergent series so strange things can happen even when you bracket them so for example this guy here the one minus one plus one minus one

Divergent series if you just bracketed right you'll actually get a convergent series with a certain sum and actually you can bracket this in many many different ways and get different sums up and maybe that's also a puzzle for you

To sort out how can you reproduce thing to get different different sums out well there's really amazing things to be discovered yet the first one is basically is saying and you can actually with what I've showed you today you can

Actually figure out the details yourself is that if I give you an arbitrary infinite series there's three different cases so you can ask in how many different sums can I rearrange this thing so the first case is no matter how

I rearrange this thing I'll never get a get a sum second case is it rearranges in all possible sums and the third case is it just as one sum no matter how you rearrange it now this gets even freakier when you considering infinite series of

Complex numbers or infinite series of vectors and you can do this in Banach space and all kinds of other things but maybe maybe that's that's enough for today

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