Not all series converge.
[MUSIC]
Let's suppose that a series converges. So let's suppose that the sum
k goes from 1 to infinity of a sub k is
equal to L. And formally what that means is that the
limit as n approaches infinity of the finite
sum k goes from 1 to n of a sub k is
equal to L.
Right to say the infinite series equals L, is
the say the limit of the partial sums is L.
I can modify this slightly without affecting the limit L.
What I mean, is that the limit as n approaches infinity of the sum
k goes from 1 to n minus 1 of a sub k is also equal to L.
Why is that?
Both of these amount of adding up a bunch of
terms in the sequence and seeing what i get close too.
Let me be a little bit more precise to see formally why
i can conclude that these two limits are both equal to L.
I mean assuming that the series converges to L.
So i can define, the nth partial sum.
As the sum k goes from 1 to n of a
sub k. And to say that the series k
goes from 1 to infinity of a sub k has value L, is just to say that the
limit of the nth partial sum as n goes to infinity is equal to L.
All right?
This is the definition of this. Now, here's the subtle point.
The limit of S sub n equals L as n approaches infinity.
Is the same as saying that the limit of S sub
n minus 1, as n approaches infinity is equal to L.
Why is this the same as this?
Well, this is saying that, if I choose little n big enough.
I can get S sub n as close as I like to L.
Well, this is really saying the same thing, right?
If I choose n just a little bit bigger,
one bigger, then I can guarantee that S sub
n minus 1 is just as close to L.
So asserting this limit is really the same as asserting this limit.
But this statement can be rewritten using this.
So what this last statement is really saying, is that the limit as
n approaches infinity. Now, what's S of n minus 1, while it's
this up here, that saying, the sum k, goes from 1 to n minus 1 of
a sub k is equal to L.
So, I've got two sequences, both of whose limits are L.
And that means the limit of their difference is zero.
Okay.
So I've got these two sequences and they've both got
a common limit of L, so I'm going to take their difference.
And let's see what I get.
So if I take their difference, I get the limit
as n goes to infinity of the sum k goes
from 1 to n of a sub k. Minus the limit,
n goes to infinity, of the sum, k goes from 1 to n minus
1, of a sub k. And that's L minus
L, so this difference is 0. But I can simplify this a bit.
Al right. This is a difference of limits.
Which is the limit of
the difference.
So this is the limit as n goes to infinity of the difference of these two things.
Which is the sum, k goes from 1 to n, a sub k minus
the sum k goes from 1 to n minus 1, of a sub k.
But now, what's this?
Well, this is really the limit as n goes from infinity of this sum is a sub 1 plus
a sub 2 plus dot, dot, dot plus a sub n minus.
What am I subtracting here?
I'm subtracting a sub 1 plus a sub 2 plus dot, dot, dot plus a sub n minus 1.
So if I add up a sub 1 through a sub
n and then I subtract everything except for a sub n.
What I'm really taking the limit of is just a sub n.
So this is the limit as n goes to infinity of just a sub n by itself.
And what we've said here is that that limit is 0.
So the limit of the nth term is 0.
What have we proved?
So the conclusion was that the limit of the nth term is 0.
The assumption at the beginning was that the series converged to L.
So what we've really shown is the following.
We've shown that if this series converges, then the limit of the nth term is 0.
So let's turn that around. Let's take the contrapositive.
So here's the original statement.
If the series converges, then the limit of the nth term is 0.
And here's the contra positive, I just turn it around.
If the limit is not 0
meaning that either the limit doesn't exist or the limit does exist.
But is some number that isn't 0 then the series has to diverge.
Because if the series did converge, then the limit would have to be 0.
What we have here is a test for divergence.
Well here's how this works.
This is the question that we're always being asked.
Question, does the series converge or diverge?
And what we can do now,
is we can take a look at the limit of the nth term.
And if that limit is not 0 then I know that the series diverges.
On the other hand, it's important to keep track
of the direction that this argument works in, right?
If it happens that the limit is equal to
0, then the series might converge, it might diverge.
In that case, this test is silent.
It doesn't tell us any information.
But if we know that the limit is not 0, then I know that the series diverges.
Let's try this on an example.
So the original question was this, does the series n goes from 1
to infinity of n over n plus 1, converge or diverge?
We'll look at the limit of the nth term. So let's
look at the limit as n goes to infinity of the nth term which is
n over n plus 1. That limit is 1 and 1 is not 0.
So the series n over n plus 1, n goes from 1 to infinity diverges.
And that hopefully makes sense because to say
that the limit of n over n plus 1 is equal to 1.
Means that this series involves adding up
numbers that eventually are very close to 1.
And if you add up a bunch of numbers that are very close to 1, well
then your almost adding up 1 plus 1 plus 1 plus 1 and that certainly diverges.
In this case, because the limit isn't 0, right?
It can't be that the series converges.
Because if the series were to converge then this limit would have to be 0.
But the limit's not 0 so the serious must diverge.
There's a lot of stuff going on here.
So if you're having some trouble keeping track of all the
moving pieces, Here's a different way to think about what's going on.
Way back to the beginning of this talk.
The very first thing we were looking at was this.
If a series converges, then, the limit of the nth term is equal to 0.
Now, starting from that premise, we then concluded this, that if the
limit of the nth term is not 0, then the series diverges.
But putting these two statements next to each
other, it can be a little bit confusing, right?
Why is diverges the conclusion of this statement but converges
is the assumption that I have to make over here?
Maybe they look like they're out of order.
Well, one way to think about this is to make it a bit more
real world. Instead of thinking about series.
Let's think about rain and clouds.
If it's a rainy day, it's then a cloudy day.
Alright, the rain has to come from somewhere.
So, raining implies cloudy. What happens if I negate these?
Alright, what happens if it's not raining?
Is it then nessecarily not cloudy? No, that's not true, there's plenty
of days when there's no rain but there's still clouds in the sky.
What if I turn this implication here around, right, is this a true statement?
Yes. If it's not cloudy then it's not raining.
Because if it were raining it has to be cloudy.
So it's that same kind of thinking now,
that I want to apply to this statement about series.
I'm starting with the statement that converges implies
the limit of the nth term is equal to 0, right?
I'm starting with the statement, like rain implies clouds.
And now I'm want to turn it around.
So I'm want to say not converges and not the limit of the nth term is 0.
But then it's going to have to also reverse the implication arrow.
It's called the contrapositive.
So what is this statement saying?
It's saying that if it's not the case that the limit of the nth term is 0,
which I could write this way. Then the series doesn't converge.
Which is just another way to say it diverges.
And that's the statement that I ended with, right?
I'm ending with the statement that if the limit
of the nth term isn't 0, then the series diverges.
But it's super important to keep track of the direction of this relationship.
This statement, diverges implies the limit of the nth term isn't 0, that thing's not
true, right? Just like not rainy implies not cloudy
isn't a true statement. But this statement is
true, if the limit isn't 0,
then, the series diverges.