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Let's look at ratios.

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[SOUND]

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We've already seen how to determine the
convergence of a geometric series.

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For example, does the series the sum n
goes from 1 to infinity

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of 1 over 4 to the n converge or

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diverge?
Well this series converges, right.

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By which test?
By the geometric series

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test, if you like.

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Because the common ratio, all right, this
common ratio between

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the terms is 1 4th And that's less than 1.

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What if we modify the series somewhat?
So what if instead of looking at this

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series, we looked at the series the sum n
goes from 1 to infinity out of 1

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over 4 to the n, but say n to the 5th over
4 to the n.

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That series is not a geometric series
anymore, but

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it ends up at least approximately looking
like one.

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Let me just give some names to these
things.

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Let's say that a sub n is n to the 5th
over 4 to the n.

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And what I'm trying to determine is the
sum n goes from 1 to infinity of a sub n.

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And I'm probably not going to actually
determine this value right.

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But I'm just trying to answer

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the question of whether or not that series
converges or diverges.

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Now, if I plug in really big values for n,

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I can get a sense of what these terms look
like.

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All right.

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A sub 1000 is well, hard to write down,
but

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I can at least write down 1000 to the 5th.

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Now if this were actually a geometric
series, right,

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the ratio between subsequent terms would
be the same.

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This isn't a geometric series, but let's
see if we're close.

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So a sub 1001 divided by a sub 1000.

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Well a sub 1001 is 1001 of the 5th power
over 4 to the 1001st

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power divided by a sub 1000, which is 1000
to the 5th power divided

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by 4 to the 1000th power.
I could rewrite this, right?

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because I've got a fraction with fractions
in the numerator and the denominator.

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I could rewrite this as 1001 to the 5th
times 4

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to the 1000th power, divided by 1000 to
the 5th times.

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And the denominator and the numerator,
write that

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in the denominator, is 4 to the 1001st
power.

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Now I should split this up and think about
these two pieces separately.

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Right?

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How big are, are these two pieces?

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Well, 1001 to the 5th power divided by
1000 to

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the 5th power, this piece here is pretty
close to 1.

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And this piece here, is, well it's exactly
1 4th, so if I'm got a number

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that's close to 1 and I'm multiplying it
by a number that's close to 1 4th,

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this fraction is well, it's not equal to,
but it's about 1 4th.

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And if you think about it, there's nothing

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special about 1000 here, except that it's
really big.

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So the ratio between subsequent terms, at
least if those those terms are

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far enough out in the sequence, that ratio
is close to 1 4th.

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Can I make this precise?
I'll use a limit,

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what I'm saying is that the limit as n
approaches infinity of a

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sub n plus 1 over a sub n, in this case is
equal to a quarter.

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That if I choose n big enough, I can make
that ratio close.

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How close?
As close as you want to a quarter.

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I can be very explicit in this case.

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I, I could pick an explicit value of
epsilon.

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Let's just

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pick 0.1.

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And then it turns out that whenever little
n is bigger than 15, a sub n plus 1

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over a sub n is, in fact, within epsilon
of my limiting value of a quarter.

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And in particular if a sub n plus 1 over a
sub n, is within 0.1 of a quarter.

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Well that also means that a sub n plus 1
over a sub n is

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less than a quarter plus 0.1.

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And that's enough to set up a comparison
with a geometric series.

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I could multiply this inequality by the
positive number a sub n.

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And I'd find out that a sub n plus 1 is
less than a quarter plus

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0.1 times a sub n.
Well, this also means that a sub n plus 2

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is less than a quarter plus 0.1 times a
sub n plus 1.

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But a sub n plus 1 is less than a quarter
plus 0.1 of a sub n,

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so a sub n plus 2 is less than a quarter
plus 0.1 squared

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times a sub n.
Now, this fact with n replaced

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by n plus 2, tells me that a sub n plus 3
is

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less than a quarter plus 0.1 times a sub n
plus 2.

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But a sub n plus 2 is less than a quarter
plus 0.1 squared times a sub n,

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so a sub n plus 3 is less than a quarter
plus 0.1 cubed times a sub n.

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Well, this works in general, right.

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What I'm really showing here is, is what?
I'm showing that a sub n

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plus k is less than a quarter plus 0.1 to
the kth power

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times a sub n, and this is true whenever n
is at least 15.

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So in particular, I'm showing that a sub
15 plus k.

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Is less than a quarter plus point 1 to the
kth power times a sub 15.

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The key

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point here is that a quarter plus epsilon
is still less than 1.

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Well, the series that we're interested in
is the sum

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n goes from 1 to infinity of a sub n.

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And that series converges precisely when
the series n

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goes from 15 to infinity of a sub n
converges.

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The first 14 terms here that aren't
included there don't effect convergence

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at all.
All right?

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That's just adding 14 more numbers to
this.

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But now how do we analyze this?

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Well here's the deal right, I've got a
bound on the size of these

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a sub n's as long as n is at least 15, I
got this.

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So the sum n goes from 15 to infinity of a
sub n

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is equal to the sum k goes from 0 to
infinity of a sub 15 plus k.

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Right, this 0 term here is just a sub 15,

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which is exactly the same as the first
term here.

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K equals 1 is a sub 16 here, and that's
exactly the next term here.

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So these are really the same series.

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But I, what do I know about this series?
Well that series converges, why?

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Because the sum k goes from 0 to infinity
of a

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quarter plus 0.1 to the k times a sub 15
converges.

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Why does this series convergence affect
this series convergence?

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Well, this series's terms are bigger than
this series terms.

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In this series, terms are all positive.

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So this follows from the comparison test,
that

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if this series converges, then this series
converges.

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But now why does this series converge?

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Well, this series converges by, say, the
geometric

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series test.

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And how does the geometric series test
effect this?

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Well, this is a geometric series.
This a sub 15 is just a constant, and the

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common ratio is a quarter plus 0.1, and a
quarter plus 0.1 is less than 1.

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So the common ratio, being less than

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1, means that this geometric series
converges.

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Means that this series converges.

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Means that the original series that we're

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studying converges.

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This is a specific case of a very general
procedure.

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That I'm studying the convergence of some
series the sum

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n goes from 1 to infinity of a sub n.

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All right, I want to know if that
converges or diverges, and let's

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suppose that all the terms in the sequence
a sub n are positive.

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What I'm suggesting that we try to do is
to look at the limit as

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n approaches infinity of a sub n plus 1
over a sub n.

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All right.

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This series is almost surely not a
geometric series, right?

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Most series aren't geometric series, but I
could try

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to see, is it close to a geometric series?

131
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What is the ratio between subsequent
terms?

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Is it close to anything eventually, right?

133
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That's what the limit's asking me.

134
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So let's suppose that this limit does
exist, let's suppose that

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this limit equals L, and let's suppose
that L is less

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than one.

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Well, then what I'm suggesting is that we
pick some small epsilon.

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how small?

139
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Well, I went epsilon so small that L plus
epsilon is less than 1.

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If L is less than 1, I can definitely pick
epsilon small enough,

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so that even if I add epsilon to L, I'm
still less than 1.

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I'm going to use epsilon in this limit,
all right.

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The fact that this limit equals

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L, means that there's some big N, let's
just say, find big N.

145
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So that whenever little n is bigger than
or equal to big N, a

146
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sub n plus 1 over a sub n, should be
within epsilon of L.

147
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And, in particular, this should be less
than L plus epsilon.

148
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And why does that help?
Well, this inequality implies

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that a sub big N plus little k is less
than L plus

150
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epsilon to little k times a sub big N.
Right, where does this come from?

151
00:10:21,310 --> 00:10:24,410
Well it really does come from just
applying this a ton of times.

152
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What this is saying is that as long as n
is at least big N, then the ratio between

153
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subsequent terms is less than this factor.
And that means that I can

154
00:10:35,630 --> 00:10:40,080
overestimate the big N plus kth term by
this factor,

155
00:10:40,080 --> 00:10:44,130
the kth power times, just whatever the big
Nth term is.

156
00:10:44,130 --> 00:10:45,170
Now, how do I use this?

157
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Well, the sum, k goes from 0 to infinity
of this, of L plus epsilon to the k

158
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times a sub N converges, and why is that?
Well this is a geometric series with

159
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common ratio L plus epsilon.
But that common ratio is less than 1.

160
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All right, that's exactly why I picked
epsilon so small up here.

161
00:11:11,860 --> 00:11:13,760
So I've got a geometric series with common

162
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ratio less than 1, so this series
converges.

163
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But then, by the comparison test, I know
something about this series.

164
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I know that the series little n equals big

165
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N to infinity of a sub n also converges

166
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because these terms are all over estimated
by these terms.

167
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And consequently, the original series, the
sum n goes from 1 to

168
00:11:41,200 --> 00:11:46,200
infinity of a sub n, also converges.
Because I

169
00:11:46,200 --> 00:11:51,510
can get this from this, just by adding on
finitely many extra terms.

170
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We can summarize this.

171
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So here's a statement of what's normally
called the ratio test.

172
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And it goes like this.

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I want to know whether the series, n goes
from one to infinity of a sub n,

174
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converges or diverges.
And I'm only going to study series,

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00:12:12,060 --> 00:12:16,840
the terms of which are all positive.
And I need that because my argument

176
00:12:16,840 --> 00:12:19,380
was using the comparison test, and the
comparison

177
00:12:19,380 --> 00:12:23,700
test requires series with at least
non-negative terms.

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All right.

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00:12:23,970 --> 00:12:28,810
Then I'm going to look at the limit n goes
to infinity of a sub

180
00:12:28,810 --> 00:12:33,400
n plus 1 over a sub n, the limit of the
ratio of subsequent terms.

181
00:12:33,400 --> 00:12:35,350
Let's suppose that that limit exists

182
00:12:35,350 --> 00:12:38,390
and equals L, then there's three
possibilities.

183
00:12:39,500 --> 00:12:43,010
If that limit is less than 1,

184
00:12:43,010 --> 00:12:48,070
then this series converges and it's
converging because I can do a

185
00:12:48,070 --> 00:12:53,229
comparison test between this series and a
convergent geometric series.

186
00:12:55,300 --> 00:13:02,200
If that limit is bigger than 1, then this
series diverges.

187
00:13:02,200 --> 00:13:04,350
And I know that because, again, if this
limit is

188
00:13:04,350 --> 00:13:08,110
bigger than 1, I can compare, maybe not
the whole series.

189
00:13:08,110 --> 00:13:13,710
But at least after a while, for some, past
some big nth term, I can compare this

190
00:13:13,710 --> 00:13:21,130
series to a divergent geometric series.
And then finally, maybe L is equal

191
00:13:21,130 --> 00:13:21,510
to 1.

192
00:13:21,510 --> 00:13:25,180
And in that case, the ratio test is
silent.

193
00:13:25,180 --> 00:13:27,205
It doesn't tell us any information either
way.

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[SOUND]

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00:13:34,967 --> 00:13:38,572
[SOUND]