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Not all series converge.

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[NOISE]

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Let's

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suppose that a series converges.
So let's suppose that the sum

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k goes from 1 to infinity of a sub k is

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equal to L.
And formally what that means is that the

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limit as n approaches infinity of the
finite

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sum k goes from 1 to n of a sub k is

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equal to L.

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Right to say the infinite series equals L,
is

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the say the limit of the partial sums is
L.

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I can modify this slightly without
affecting the limit L.

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What I mean, is that the limit as n
approaches infinity of the sum

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k goes from 1 to n minus 1 of a sub k is
also equal to L.

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Why is that?

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Both of these amount of adding up a bunch
of

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terms in the sequence and seeing what i
get close too.

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Let me be a little bit more precise to see
formally why

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i can conclude that these two limits are
both equal to L.

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I mean assuming that the series converges
to L.

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So i can define, the nth partial sum.

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As the sum k goes from 1 to n of a

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sub k.
And to say that the series k

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goes from 1 to infinity of a sub k has
value L, is just to say that the

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limit of the nth partial sum as n goes to
infinity is equal to L.

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All right?

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This is the definition of this.
Now, here's the subtle point.

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The limit of S sub n equals L as n
approaches infinity.

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Is the same as saying that the limit of S
sub

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n minus 1, as n approaches infinity is
equal to L.

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Why is this the same as this?

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Well, this is saying that, if I choose
little n big enough.

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I can get S sub n as close as I like to L.

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Well, this is really saying the same
thing, right?

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If I choose n just a little bit bigger,

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one bigger, then I can guarantee that S
sub

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n minus 1 is just as close to L.

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So asserting this limit is really the same
as asserting this limit.

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But this statement can be rewritten using
this.

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So what this last statement is really
saying, is that the limit as

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n approaches infinity.
Now, what's S of n minus 1, while it's

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this up here, that saying, the sum k, goes
from 1 to n minus 1 of

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a sub k is equal to L.

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So, I've got two sequences, both of whose
limits are L.

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And that means the limit of their
difference is zero.

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Okay.

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So I've got these two sequences and
they've both got

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a common limit of L, so I'm going to take
their difference.

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And let's see what I get.

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So if I take their difference, I get the
limit

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as n goes to infinity of the sum k goes

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from 1 to n of a sub k.
Minus the limit,

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n goes to infinity, of the sum, k goes
from 1 to n minus

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1, of a sub k.
And that's L minus

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L, so this difference is 0.
But I can simplify this a bit.

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Al right.
This is a difference of limits.

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Which is the limit of

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the difference.

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So this is the limit as n goes to infinity
of the difference of these two things.

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Which is the sum, k goes from 1 to n, a
sub k minus

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the sum k goes from 1 to n minus 1, of a
sub k.

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But now, what's this?

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Well, this is really the limit as n goes
from infinity of this sum is a sub 1 plus

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a sub 2 plus dot, dot, dot plus a sub n
minus.

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What am I subtracting here?

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I'm subtracting a sub 1 plus a sub 2 plus
dot, dot, dot plus a sub n minus 1.

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So if I add up a sub 1 through a sub

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n and then I subtract everything except
for a sub n.

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What I'm really taking the limit of is
just a sub n.

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So this is the limit as n goes to infinity
of just a sub n by itself.

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And what we've said here is that that
limit is 0.

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So the limit of the nth term is 0.

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What have we proved?

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So the conclusion was that the limit of
the nth term is 0.

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The assumption at the beginning was that
the series converged to L.

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So what we've really shown is the
following.

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We've shown that if this series converges,
then the limit of the nth term is 0.

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So let's turn that around.
Let's take the contrapositive.

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So here's the original statement.

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If the series converges, then the limit of
the nth term is 0.

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And here's the contra positive, I just
turn it around.

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If the limit is not 0

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meaning that either the limit doesn't
exist or the limit does exist.

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But is some number that isn't 0 then the
series has to diverge.

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Because if the series did converge, then
the limit would have to be 0.

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What we have here is a test for
divergence.

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Well here's how this works.

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This is the question that we're always
being asked.

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Question, does the series converge or
diverge?

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And what we can do now,

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is we can take a look at the limit of the
nth term.

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And if that limit is not 0 then I know
that the series diverges.

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On the other hand, it's important to keep
track

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of the direction that this argument works
in, right?

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If it happens that the limit is equal to

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0, then the series might converge, it
might diverge.

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In that case, this test is silent.

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It doesn't tell us any information.

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But if we know that the limit is not 0,
then I know that the series diverges.

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Let's try this on an example.

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So the original question was this, does
the series n goes from 1

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to infinity of n over n plus 1, converge
or diverge?

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We'll look at the limit of the nth term.
So let's

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look at the limit as n goes to infinity of
the nth term which is

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n over n plus 1.
That limit is 1 and 1 is not 0.

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So the series n over n plus 1, n goes from
1 to infinity diverges.

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And that hopefully makes sense because to
say

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that the limit of n over n plus 1 is equal
to 1.

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Means that this series involves adding up

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numbers that eventually are very close to
1.

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And if you add up a bunch of numbers that
are very close to 1, well

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then your almost adding up 1 plus 1 plus 1
plus 1 and that certainly diverges.

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In this case, because the limit isn't 0,
right?

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It can't be that the series converges.

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Because if the series were to converge
then this limit would have to be 0.

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But the limit's not 0 so the serious must
diverge.

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There's a lot of stuff going on here.

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So if you're having some trouble keeping
track of all the

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moving pieces, Here's a different way to
think about what's going on.

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Way back to the beginning of this talk.

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The very first thing we were looking at
was this.

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If a series converges, then, the limit of
the nth term is equal to 0.

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Now, starting from that premise, we then
concluded this, that if the

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limit of the nth term is not 0, then the
series diverges.

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But putting these two statements next to
each

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other, it can be a little bit confusing,
right?

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Why is diverges the conclusion of this
statement but converges

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is the assumption that I have to make over
here?

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Maybe they look like they're out of order.

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Well, one way to think about this is to
make it a bit more

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real world.
Instead of thinking about series.

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Let's think about rain and clouds.

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If it's a rainy day, it's then a cloudy
day.

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Alright, the rain has to come from
somewhere.

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So, raining implies cloudy.
What happens if I negate these?

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Alright, what happens if it's not raining?

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Is it then nessecarily not cloudy?
No, that's not true, there's plenty

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of days when there's no rain but there's
still clouds in the sky.

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What if I turn this implication here
around, right, is this a true statement?

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Yes.
If it's not cloudy then it's not raining.

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Because if it were raining it has to be
cloudy.

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So it's that same kind of thinking now,

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that I want to apply to this statement
about series.

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I'm starting with the statement that
converges implies

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the limit of the nth term is equal to 0,
right?

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I'm starting with the statement, like rain
implies clouds.

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And now I'm want to turn it around.

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So I'm want to say not converges and not
the limit of the nth term is 0.

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But then it's going to have to also
reverse the implication arrow.

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It's called the contrapositive.

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So what is this statement saying?

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It's saying that if it's not the case that
the limit of the nth term is 0,

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which I could write this way.
Then the series doesn't converge.

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Which is just another way to say it
diverges.

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And that's the statement that I ended
with, right?

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I'm ending with the statement that if the
limit

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of the nth term isn't 0, then the series
diverges.

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But it's super important to keep track of
the direction of this relationship.

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This statement, diverges implies the limit
of the nth term isn't 0, that thing's not

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true, right?
Just like not rainy implies not cloudy

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isn't a true statement.
But this statement is

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true, if the limit isn't 0,

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then, the series diverges.