Hi, everyone!
Welcome to this third video
in my series about Ramanujan
Now, last time I did promise you that this video
would be a little bit controversial.
Some of you may have even guessed what I'm going to talk about today.
It's a result that Ramanujan
first discovered while still in India,
before he came to Cambridge,
but only something that he perfected
with the help of G.H. Hardy.
It's a result about
the sum of all integers.
So it's 1 plus 2 plus 3 plus 4,
going on forever.
Now, common sense tells us that this
is going to go off
to infinity.
But Ramanujan had a different answer.
And his answer does make sense, from the correct
point of view.
But you're not going to like it!
Ramanujan's answer
for the sum of all integers
was - 1/12.
Which is crazy, right? It sounds crazy, right?
If you don't like it, I don't blame you
if you don't like it.
Even Ramanujan thought
it was crazy.
He wrote letters to G.H. Hardy
sending him this result,
but saying "Look, please don't send me to the lunatic asylum."
So this does sound crazy,
especially if you compare it with the classic formula
of adding up integers 1 to n.
So this is a finite sum, where you've got
1 plus 2 plus 3, up to n.
This is called a triangle number,
if you can imagine having
a row of one, a row of two, a row of three,
a row of four--it would make a triangle shape.
Now, there is a formula for working out a triangle number.
If you consider taking the first number,
then adding it to the last number,
you get n plus 1.
If you take the second number,
and add it to the second-to-last number,
you get n plus 1 again.
In fact, if you keep adding up in pairs
in that way,
each pair
is n plus 1.
And how many pairs do we have?
Well, we've got n over 2 pairs.
So all together, the sum
is n over 2 lots of n plus 1.
And you get this formula for triangle numbers.
And you can see that the triangle numbers are getting
bigger and bigger and bigger.
What Ramanujan was doing
was essentially considering the infinite
triangle number,
and he was getting this strange answer--
-1/12.
The thing is,
Ramanujan hadn't been the first
mathematician to notice that answer.
Other mathematicians had seen it, as well.
Euler had seen that answer.
Riemann had seen that answer.
What was new
was Ramanujan's method
for getting that answer.
And that's what I want to explain in this video.
And I'm hoping that by explaining it,
you'll have a better appreciation
of where that answer comes from.
But I'll tell you right now,
up front,
what I want you to take away from this video.
So what Ramanujan had done,
his method, could be applied
to traditional sums.
And if you apply it to traditional sums,
you would get the answer you expect.
But then, if you applied Ramanujan's method
to, say, the sum of all integers,
then you get this -1/12.
So what Ramanujan had done
is he had generalized the idea of a sum.
Ramanujan's method was based on
estimating the area under a curve.
So you can estimate
the area under a curve
by using rectangles.
Each rectangle has a width of just one unit,
and the height of the rectangle
is given by the value
of the function at integers.
So the total area of those rectangles
is just the sum of the function at integers,
up to some integer, n.
Now, there's going to be a difference between
that value and the true area
under the curve.
But there is a formula for that difference.
It's called the Euler-Maclaurin formula,
and it tells us
that the difference between the sum
and the integral
is given by
repeatedly differentiating
the function
and then evaluating that at n,
minus evaluating that at zero.
Now, these terms, labeled B,
are called Bernoulli numbers.
The sequence for Bernoulli numbers
looks like this:
and at the very end
we have this small remainder term
and that is small enough that
we can ignore it.
If you take all the terms that don't depend on n,
you get something that Ramanujan called
"the constant of the function."
So this is a constant.
This will always stay the same
in our Euler--Maclaurin formula,
a n tends off to infinity.
Edward Frenkel, on Numberphile,
gave this a lovely description.
He described this constant as being like
a 'golden nugget,' surrounded by infinity.
Now, G.H. Hardy pointed out
to Ramanujan
that if you want to use this method to find sums,
then when that function f is a convergent function,
then this should equal the traditional sum.
So Hardy modified the idea,
and he gave this as the constant instead.
So this is still a constant,
but it does depend on your choice of (a).
And (a) is usually zero
or infinity.
If f is convergent, then you would use infinity.
And in that Euler--Maclaurin formula,
a lot of terms actually disappear--
because f is convergent
it's tending to zero.
So a lot of terms disappear,
and you'll find that
that modified constant
is equal to the traditional sum.
If f was divergent,
then that wouldn't be defined.
So instead, we would evaluate the constant at zero,
and in that case, you would get Ramanujan's constant.
So let's do it!
Let's find the constant
for the sum of 1 to infinity.
In that case,
the function is just x.
And if we apply Ramanujan's constant to that,
you will get the same answer that he did.
You will find that it is equal to
-1/12.
Now, this isn't the only method
that comes up with that answer--
the -1/12 answer.
For example,
you can use the Riemann zeta function
and that will also give you
-1/12.
So this isn't just nonsense.
And today, scientists use this in high level physics.
Apparently, they use it in string theory.
But don't ask me about that--
it sounds hard.
If you're still not happy,
then think about it some more.
But if it helps,
let me say this:
Maths is about solving problems
through abstraction.
Like those triangle numbers we looked at earlier,
those are finite sums,
but we can work those out instantly,
because we had a formula
for working out triangle numbers.
So what we did there
is turn that problem
into a more abstract problem.
Well, if we take that one step further,
we can work out infinite sums, as well.
So convergent infinite sums
are those where you are getting closer and closer
to a value.
And so we take that value to be the sum
of the series.
And they act a lot like finite sums.
You can add them and multiply them
like you can with finite sums.
And you might feel comfortable with those kind of sums already.
Maybe you've seen them before,
maybe at school.
So you're feeling OK with those.
But they do come with their own
infinite weirdness themselves.
For example, in a previous video,
we looked at Leibniz's pi formula.
What you've got there is
you're adding up rational numbers.
If you add up two rational numbers
you get a rational number.
So if we keep adding up rational numbers,
then we should get
a rational number.
But with this formula,
we get pi over here--we get
an irrational number.
which is really strange.
It turns out
that if consider the sequence of partial sums,
then the final answer
does not have to be a member
of that sequence.
So infinite weirdness exists already.
If we just take this one step further,
one step more abstract,
then we can work out values
for all kinds of infinite sums,
even those that aren't getting
closer and closer to a value.
But the important thing is,
is any method we have
has to be able to be applied downards.
So we have to be able to apply that method
to, say, convergent sums,
and still get the expected answer.
You just have to embrace
the infinite weirdness.
This was my third video about Ramanujan.
inspired by the release of the new film,
"The Man Who Knew Infinity,"
all about the life of Ramanujan.
I might come back with another video about the film itself,
but until then,
if you have been,
thanks for watching.