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The ratio test.

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[SOUND]

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The ratio test asks us to look at the
limit of the ratio of subsequent terms.

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Well, here's a precise statement.

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So the question is whether the sum n goes
from

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one to infinity of a sub n converges or
diverges.

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I'm going to want to assume that all of
the terms that I'm adding up, all of

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the a sub n's, are positive, in order that
I can use the comparison tests.

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I'll compute the limit of the ratio of
subsequent terms.

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So I'm setting big L to be the limit as n

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approaches infinity of a sub n plus one
over a sub n.

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And then there's three possibilities.

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Maybe that limit, maybe big L is less than
one.

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And in that case, the ratio test says that
the series converges.

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Maybe that limit is bigger than one.

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In which case the ratio test says that the
series diverges.

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And maybe that limit is equal to one.

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In which case, the limit test is simply
silent.

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It might converge, it might diverge.

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We just don't know.

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The ratio test doesn't tell us any
information when L is equal to one.

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Why does this work?

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What if the limit is less than one?
So I'm trying to figure out what happens

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when big L is less than one.

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I want to show the series in that case
converges.

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Well, if big L is less than one, I can
pick some very tiny

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value of epsilon so that even big L plus
epsilon is less than one.

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Then I can use epsilon in the definition
of limit.

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So there must be some big N so that
whenever little n is at least

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big N, the thing I'm taking the limit of,
a sub n plus one over a

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sub n must be within epsilon of L.

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So in particular, a sub n plus one over a
sub n must be less than L plus epsilon.

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Now how is this fact helpful?
Well, I'm trying to calculate this series.

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I'm trying to sum a-sub-n, n goes from one
to infinity.

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But I can break it up into two pieces,
when n is

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smaller than big N, and when n is bigger
than big N.

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Well, here's the piece when n is smaller

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than big N.

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It's just the sum of you know, a finite
sum.

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I'm not going to have any trouble adding
those numbers up.

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But then I've got the rest of the series
that I have to add up.

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Right, I have to add up a sub big N, plus
a sub big N plus one, and so on.

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And this is the hard part, right?

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Because I've got dot dot dot, forever.
But what I know now is that I can

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control the ratio between subsequent
terms, maybe not here, but at least here.

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I know that a sub big N plus one is less
than R times a sub big N.

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And I know that a sub big N plus two is
less than r times a sub big

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N plus one which itself is less than r
times a sub big N.

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So, all of that is to say that this thing
here is well, not equal to this,

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but this is an overestimate of what was
there before.

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So now I've got this series of at least

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less of than this first part, plus the
rest.

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But what do I know about the rest?

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Well the rest of this thing is now a
geometric series and

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I know this geometric series converges
because little r is less than one.

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So I can replace that with this convergent
geometric series.

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And in light of this, all right, this is
really now setting up a comparison test.

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I am really saying that the sequence of
partial sums is bounded by this.

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And since the sequence of partial sums is
a monotone, because all the terms

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that I'm adding up are positive, I then
know that this series converges.

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And what if the limit is bigger than one?
So I am trying to figure

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out what happens when big O is at least
one and

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I want to show that in that case the
series diverges, or big

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O is bigger than one then I can choose
some tiny

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epsilon so that big O minus epsilon is
also bigger than one.

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Then I can use that epsilon in the
definition of limit.

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So there's some big N so that whenever
little n is bigger

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than or equal to big N, this ratio, the
thing I'm taking

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the limit of, is within epsilon of the
limit L.

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So in particular, this ratio is bigger
than big L minus epsilon.

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Why is that significant?

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What I'm trying to calculate is this
series.

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Or at least I'm trying to figure out if it
converges or diverges.

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And I can start just adding up terms,
right?

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A sub one plus a sub two.

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Eventually I get up to a sub big N minus
one,

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then a sub big N plus a sub big N plus
one,

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and so on.

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I keep on adding up terms, making a

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limit of partial sums, you know, strictly
speaking.

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But I know that all of these terms are
positive.

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So, if I throw away a bunch of terms,
well, this isn't equal to this anymore,

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that this is now larger than this thing

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where I've thrown away a whole bunch of
terms.

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We're going to start with a sub big N
term.

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Now, what else do I know?

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Because a sub

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little n plus one over a sub n.

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Is bigger than this whenever little n is
bigger than or equal to big N.

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I can underestimate this.

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Alright, this is saying that a sub big N
plus

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one is larger than r times a sub big N.

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Telling me that this, is bigger than r
times this.

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Which is also bigger than r times this.

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Which means this is larger than r squared
times this.

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This term here, a sub big N plus three is
larger than r cubed times this.

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What I'm really saying is that this

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infinite series is at least this infinite
series.

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But what kind of infinite series is this?

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Well this is just a geometric series,
right?

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That's the geometric series.

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The sum k goes from zero to infinity of r
to the k times a sub big n.

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And what do I know about that infinite
series?

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That infinite series diverges, right?

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I can make the partial sums for this as
large as I'd like.

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As long as I go far enough out in the
sequence.

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Well, that means that this series also
diverges.

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And what does the ratio test say if the
limit is equal to one?

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When L is equal to one, the series might
converge or it might diverge.

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The ratio test is just silent in that
case.

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It doesn't tell us any information.
And, and why not?

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Well, when L was less than one, then I
could pick a value

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of epsilon so small that even L plus
epsilon was less than one.

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And, when L was bigger than one, I could
pick a value of

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epsilon so small that even big L minus
epsilon was bigger than one.

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But when L is equal to one, there's no
hope for me being

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able to choose a small enough epsilon so
that I can control the ratio between

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subsequent terms, to eventually all be
less than

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one, or eventually all be bigger than one.

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So I'm not going to succeed in comparing
my series to a geometric series.

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So that's the ratio test.

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But finally, a warning.
The ratio test tells us to compute this,

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the limit of the ratio between subsequent
terms.

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It's important to emphasize that that
limit is not calculating this.

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What the ratio test is telling us is not
the value of this series.

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What the ratio test is providing
information about is whether or not this

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series converges or diverges.
That's what this big L is helping us find.

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[SOUND]