So today I want to try
and bend your brains a little bit today. And I'm hoping it will cause a
little bit of debate on the comments, because I know
YouTube's the home of rational and informed debate. So I look forward to that. The question is what
is this equal to? It's quite a simple sum. It starts with 1. Then I'm going to subtract 1. Then I'm going to add 1 again,
then subtract 1, then add 1, then subtract 1, then add
1, then subtract 1. And I'm going to do
this forever. You get the idea of
that, I hope. So what does that equal to? So one of the answers that it
might be is if I put the brackets like this-- here and here and
here and here-- you can see each bracket is
1 minus 1 plus 1 minus 1 plus 1 minus 1. Each bracket is 0. So you're getting 0 plus 0
plus 0 plus 0 forever. So that's going to be equal
to 0, isn't it? That's one of the answers
it could be. The problem is there
is another answer. If I do it again, we could put
the brackets here, like this. Now let's say this is-- plus again there plus
this bracket. So I started with 1 plus minus
1 plus 1-- that's a 0-- plus minus 1 plus 1. That's a 0. Et cetera, et cetera. All the brackets are 0. So all the brackets
add up to 0. But I've got a 1 at the start. So now this is equal to 1. I've got two answers. I've got 0 if I put
the brackets here. I've got 1 if I put brackets
in a different place. There is a third answer
as well, and this is the very weird one. Let's say it has a number, so
let's call it S. We're going to try and find out what
S is equal to. That's what we want to know. Let's do 1 minus S. So it's
1 minus this infinite sum. Let's do that. So let's write it out. Plus 1 minus 1 plus
1 minus 1-- right. If we take the bracket away,
this minus number will mean that all the signs will get
flipped, so you'll get 1 minus 1 plus 1 minus 1
plus 1 minus 1. That's what happens when I
take away the bracket. But what I end up with is the
thing I started with. That's just the alternating
plus and minus 1. So I've got S again. So I've got 1 minus S is
equal to S. That's OK. That's fine. You can solve that. In other words, if I take the S
to the other side, I've got 2S equals 1, which then
you can see that S is equal to 1/2. That's a weird answer. I've got a 1/2. The sum of adding
plus and minus 1 forever give you a 1/2. Well, it might be 1. It might be 0. But it might be a 1/2. So the guy who came up with
this idea was an Italian mathematician called Grandi. He did this in 1703. He was a monk. He was a mathematician. He was one of those types. And he published this. And he said this is weird. It's 0. It's 1. It's 1/2. What's that all about? And the mathematical community
had a look at it. And they said well, it
can't be 1/2, can it? I mean, you've got 1s and 0s. That's madness. It's can't be. Oh. Hang on. Oh, that's actually
quite convincing. It might be 1/2. So there was a debate about
this for a long time-- I think 150 years-- quite a debate until the 19th
century, when all this stuff with infinite sums really
got sorted out. A lot of people think that
the best answer is 1/2. I want to try and show you
why they think the best answer is 1/2. And then the one after that, I'm
going to show you one more thing to completely
bend your brain. If we pick a nice infinite sum--
because there are nice infinite sums, and there are bad
infinite sums-- one of the nice ones is this. 1 plus 1/2 plus 1/4 plus
1/8 plus 1/16. And the way you can work out
the answer for that-- actually I'm going to show you
the proper way to do it. The proper way to do is look
at the partial sums. We're going to add this
sum term by term. So let's just make a
sequence of them. I start with 1. I'll write that down. What do I get if I add
the first two terms? It's 1 plus 1/2. It's actually 3/2. If you prefer, that's 1.5. Let's add the first three
numbers together. So 1/2 plus 1/4. Let's do that. 7/4-- it's 1.75. If I add the first
four together-- 15 over 8, which is 1.875. And if you did the next lot,
you get 63 over 31-- 1.96875. You might be able to see,
they're getting closer and closer to the value 2. In general, if I picked one
in general, it would be 2 minus 1 over n. And if you can see, as the n
gets bigger, this gets tiny and disappears, and you're
just left with 2. And mathematicians are justified
in saying that the whole infinite sum
is equal to 2. If we try with Grandi's series,
it doesn't work. Look at the partial sums. The first one is 1. And you add the first
two together, you're going to get 0. You add the first three
together, you get 1 again. You add the first four together,
you get back to 0. And it keeps alternating
between 1s and 0s. And it's not getting
closer to a value. So this doesn't work with
Grandi's series. So I'm going to show
you a second method to work out sums. I'm going to take the partial
sums, and I'm going to look at the averages. I'm just going to average
as I go along. Almost the same way. I'll do it with this one first
to show you the idea. Let's take the first one. That's 1. I'm going to add the first two
partial sums together. So 1 plus 1.5, but
I'll average it. I'm going to divide by 2. So it's going to be 1 plus 3/2
and then average it like that. Average is actually
equal to 5/4. If I took the first three and
averaged them, I would have 1 plus 3/2 plus 7/4 divide by 3. And that gives me 17
over 12, and-- well, hopefully, you get
the idea of that. Again, the numbers are tending
closer and closer to 2. It's just another method
to get the same answer. It gives me 2 again. In fact, in general, what you
get is 2 minus some junk. Oh, the joke isn't important. Look. It's junk. But this junk is
getting smaller and smaller and smaller. So you're getting 2 again. It's just another way to
find the same answer. But this method can be used
with Grandi's series. Let's try it. We're averaging the
partial sums. So those are the partial sums. We start with 1. Then if you average the first
two, you get 1 plus 0 divided by 2, which is 1/2. Take the first three, and then
divide by 3 gives me 2/3. I take the first four-- 1 plus 0 plus 1 plus 0-- divide by 4. That's another 1/2 again,
if I get that right. Take the first five-- so you might be able to see
what's going on, yeah-- divide by 5. So that's 3/5. What happens is, in general,
you keep going. In general, you'll get 1/2
followed by something like 1/2 plus 1 over 2n. There we go. Again, and so you get some junk
here that's going smaller and smaller and smaller. This is all tending
towards 1/2. So together you're zoning
in onto the number 1/2. So this is more technical than
the other version I did but it's a second way to get sums. You average the partial sums. But it works for Grandi's
series. It gives me 1/2. So what's going on? What's the difference? This second method-- it
gives you sums when there are sums to find. A limit is when you're
getting closer and closer to the value. Now Grandi's series does not
have a limit, because you're not getting closer and
closer to the value. But you have this second
way of finding a sum. It's almost like a limit, but
it's not really a limit. It's a fake limit. It's a pseudo limit. It has all the properties
of limits. It does all the same things. It's so close to being a limit,
that it turns up in calculations where you expect
limits to turn up. But the difference is you're not
getting closer and closer and closer. To really bend your brain,
try and imagine this. We're going to try to imagine
doing this in the real world. Imagine a light. We're going to turn the
light on and off. So you turn the light on. You turn the light off. Now every time, if I go along
Grandi's series, every time I see a 1, I turn the light on. Every time I see a minus 1,
I turn the light off. So you turn it on,
you turn it off. You turn it on, you
turn it off. The partial sums actually tell
you if the light is on or off. If you have a 1, that means
you just turned in on. If you have a 0, that means
you've just turned it off. You're going to start
an experiment. After one minute, you
turn the light on. After half a minute, you then
turn the light off again. After a quarter of a minute,
you turn the light on. After an 1/8 of a minute,
you turn the light off. And you're turning it on and
off, but you're doing it quicker and quicker
and quicker. So you're doing that infinitely
many times. But if we add up the time
together, 1 minute plus 1/2 minute plus 1/4 of a minute
plus 1/8 of a minute-- forever-- adds up to 2 minutes. In fact, that's that
series I did there. If you remember the video we
did about Zero's Paradox, that's not just getting closer
to two minutes, you can actually compete it and
finish the whole process in two minutes. So in two minutes time, you'll
have turned on-- on and off-- the light infinitely many
times and completed it. After two minutes, is
the light on or off? If Grandi's series is 0, that
means the light is off. If Grandi's series is 1, that
means the light is on. If Grandi's series is 1/2,
what does that mean? Is it 1/2 on, 1/2 off? Is it on and off at
the same time? What do you think? So he's given a head start. He's got 100 meter head start. And then they start the face. Now Achilles sprints 100 meters,
and he catches up to where the tortoise was. But in that time, the tortoise
has moved on.
My answer to the final question, if I really had to give one, is that the light would be off because the bulb exploded from the constantly changing current (a joke). This is abstract math, and as such there comes a point, I believe, where the math cannot logically be applicable to a real-world problem. Trying to solve it in this way rightfully leaves the person dumbfounded.
Although, it could be said that, after having been switched infinitely many times, the light switch would remain unchanged because it has not yet passed the threshold for change, i.e. halfway. This begs the inexplicable question, though, of where was the light switch before it reached halfway...