Not all series converge.
[NOISE]
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.