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Coarseness.

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[MUSIC]

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Often we've been asking for things on the
nose.

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Like in this formulation of the comparison
test.

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For all n, 0 less than equal to a sub n
less that equal to b sub n, if this

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series the sum of the b sub n converges,
then

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this series the sum of the a sub n
converges.

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When you formulate it this way, I'm asking
for this inequality to hold on the

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nose that, a sub n be between 0 and b sub
n for all n.

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But now, we're beginning to see, that we
don't need to be quite so rigid.

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Well, one thing we've seen is that
convergence

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depends only on the tail of the series.

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So, something like this, this assumption
here for all n, 0 is less than

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or equal to a sub n is less than or equal
to b sub n.

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This can be weakened.

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I can get away with just assuming this,
that for all large values

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of little n, a sub n is between 0 and b
sub n.

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And I can replace the statement that, a
sub n is less

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than b sub n, with something with the
limit of the ratios.

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I'll still make the assumption that the
terms are

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non-negative, but instead of this
inequality, I could rewrite

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that to just say that, a sub n over b sub
n, this ratio is between zero and one.

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And the point I'm thinking about this
ratio, or I could just get rid of this and

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instead of thinking about this ratio being
between zero and one, I could

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just ask that the limit of this ratio be
some finite value L.

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And at that point,

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[LAUGH]

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I've written down one direction of the
limit comparison test.

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So, we've replaced the comparison test,
which is really

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a pretty rigid statement, with something
much more flexible.

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So, yeah, the limit comparison test is a

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coarser version of just the regular old
comparison test.

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[LAUGH]

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And by coarser, I actually am claiming,
this is a sort of philosophical

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claim, that the limit comparison test is
actually better than the Comparison Test.

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And it's better for two reasons.

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Well, first of all, the limit comparison
test is

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better than just the comparison test in
practice, all right?

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Beyond any sort of theoretical advantage,
I mean the

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limit comparison test just ends up being
easier to apply.

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It's easier to verify this condition, than
it

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is to cook up some sequence of b sub n's
which satisfy

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this and the sum of the b sub n's
converges, say, right?

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This is much harder to satisfy than just
some limit statement like this.

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Well, let's see and expose an example in

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practice where, where this really is much
easier.

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So, here's an example of a series.
Let's analyze the convergence of this.

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The sum n goes from 1 to infinity, of this
fraction

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n squared plus the square root of n minus
n divided

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by n to the 4th plus n squared minus n.
Does this series converge or diverge?

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What I can do is build a sequence out of
this.

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So, I'm going to call the terms here, a
sub n, and then I'm going to

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look at this other sequence, sequence b
sub n equals 1 over n squared.

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And I'm going to compute, the limit of

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the ratio between the terms in these
sequences.

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So, what's the limit as n goes to
infinity,

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of a sub n over b sub n?
Well, dividing by 1 over n, is the same

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as just multiplying by the reciprocal.
So, I'm really asking about this limit.

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But this isn't so painful now, right?

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because now, I'm multiplying n squared
times this.

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So, this limit is this fractions limit,
this fractions, n to the

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4th plus lower order terms n squared times
n squared plus lower order

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things, and the denominator is the same,
it's just n to the

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4th plus lower order of things, in this
case n squared minus n.

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So, what's this limit?

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Well I have got, n to the 4th plus lower
order terms divided by

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n to the 4th plus lower order terms, this
limit ends up being 1.

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So, the limit comparison test applies and

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[LAUGH]

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what I know, is that this series, the sum
of the b sub n's

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converges because this is just the p
series with p equals 2.

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And that means as a result of the limit
comparison test,

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I can then conclude that this very
complicated series also converges.

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But it's not just that it's easier to
apply, right?

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There's a second reason, why this kind of

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coarse thinking is really beneficial?

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And the reason is that thinking this way
actually

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gets us closer to understanding what
convergence really means.

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Alright?
What's conversions all about?

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Well, it's certainly not about the first
million series, right?

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Convergence only depends on the tail.

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And what convergence really depends on is,

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how quickly the terms are decaying to
zero.

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So, something like the limit comparison
test, that asks you to look at the limit

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of the ratio of the terms in the series,
is really asking

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you to think about how quickly those terms
are decaying to zero.

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And that's good, right?

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That means the limit comparison test isn't
just a way to get the answers more easily.

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The limit comparison test is, is really

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showing us something about what
convergence really means.

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And that's the goal of mathematics.

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The goal of mathematics isn't just
answers.

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It's understanding.

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Well, think back to that example again,
alright?

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This example wasn't just answering a
question.

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This example is demonstrating this
philosophy, right?

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That, I'm doing more than just answering
the question of this series convergence.

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I'm saying that there's a reason, an

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understandable reason why this series
should converge.

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This series converges because it
resembles, in the sense of

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this limit between the terms in these
series, but this series

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resembles 1 over n squared.

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And this p series, where p equals 2
converges,

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it's where the limit comparison test this
series does too.

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But it's not just a calculation.

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That's not just an answer.
That's a reason,

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why something that looks

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like this ought to converge

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[SOUND].