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Alternating series test.

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[SOUND]

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Alternating series and it turns out that
there's

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a particularly nice criterion for when
they converge.

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So here's what normally goes by the name
the alternating series test so the

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setup of this is as follows I'm going to
suppose that I've got a sequence.

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The terms are decreasing and all of the
terms are positive.

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And the limit of the nth term as n goes to
infinity is 0.

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So these are the assumptions.

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And then in that case, I may conclude that
this alternating series that I build from

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the sequence a sub n by multiplying by
minus 1 to the n plus 1 power.

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This alternating series converges.

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Alternating series form a class of series
for

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which the limit test is practically the
whole story.

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So given these conditions, if the limit of
a sub n is 0, then this

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series converges.

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But we already know by just the limit test
that if the limit

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of the nth term of this series, remember,
is non-zero, then the series diverges.

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But to say that this limit is non-zero is
exactly the same

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thing as saying that this limit either
doesn't exist or is non-zero.

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So, alternating series are a situation
where

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just determining the limit of the nth
term,

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or at least the absolute value of the nth
term, tells the whole story.

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If that limit is 0, then the series
converges,

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and if that limit's non-zero, then the
series diverges.

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Now, you already know that the if the
limit

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

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So, in this case, what do I have to show?

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So this is the statement that I want to
prove.

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I start with some sequence which is
decreasing.

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All the terms are positive.

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The limit of that sequence is 0.
I want to

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conclude that this series, this
alternating series, converges.

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Well what does that even mean?

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What does it mean to show that this series
converges?

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Right, I just have to remember the
definition

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of this is really something about the
limit.

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All right, it's to say that the limit of
this partial sum, the sum k goes from 1 to

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n of minus 1 to the k plus 1 times a sub
k, this sequence of partial sums should

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converge and I'll usually denote that n
partial sum by just, s sub n.

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Well let's look more carefully at these
partial sums.

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So let's look at the first few terms of
the sequence of partial sums, right?

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What happens when I just plug in a few
values of n here?

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What do I get?
Well, when I plug in n equals 1?

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It's not too interesting, right?

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It's just the sum of the first term, which
is just a

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sub 1.

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When I plug in n equals 2, that's a little
bit more interesting, right?

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That's the sum of the first two terms,
which is a sub 1 minus a sub 2.

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Why is the minus in there?

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When I plug in k equals 2 here, I get
negative 1 to the 3rd times a sub 2.

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Negative 1 to the 3rd is minus 1.

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So that's introducing a negative sign in
front of the a sub 2.

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So this is a sub 1 minus a sub 2.
What's the third partial sum?

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Well the third partial sum is a sub 1
minus a sub 2 plus the next term in

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the series which is minus 1 to the 4th
times a sub 3, it's plus a sub 3.

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What's the fourth partial sum?

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Well that's a sub 1 minus a sub 2 plus a
sub 3 minus a sub 4.

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I mean, it's maybe we're not getting a lot
of intuition here.

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The fifth one is a sub 1 minus a sub 2
plus a sub 3 minus a sub 4 plus a sub 5.

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It's a little

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bit hard maybe to tell what's going on.
So, instead of just looking at these

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numbers, I could try to look at these
partial

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sums on a number line.
And the key fact to remember here is that

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these a sub n's are decreasing, so in
something like the fifth partial

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sum, a sub 2 is smaller than a sub 1, a
sub 3 is smaller than both of

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these still.
Right?

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That's going to help us to draw a nice
picture of

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what these partial sums look like on the
number line.

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So the first partial sum, you know?

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Who knows where it is?

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It's just a sub 1 and it's somewhere on
the number line.

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But then to get from that first partial
sum to the next partial sum to get from s

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of 1 over to s of 2, all right?
I just subtract a sub 2.

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So here I am subtracting a sub 2, and that
lands me over here

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at the second partial sum, right, because
s sub 2 is a sub 1 minus a sub 2.

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Well, how do I get to the third partial
sum?

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Where is the third partial sum?

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Well, to get to s sub 2 to s sub 3, I just
add a sub 3.

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And the key fact here is that a sub n is
decreasing

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so the amount that I'm moving to the right
with a sub 3 is

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less than the amount that I was moving to
the left with a sub 2.

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So maybe here is s sub 3, the third
partial sum.

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And then to get from s sub 3 to s sub 4, I
have to subtract a sub 4,

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but again, the sequence a sub n is
decreasing, so a sub 4 is smaller than

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a sub 3, so I'm moving less to the left
than I was

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to the right to go from s sub 2 to s sub
3.

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So now I subtract a sub 4, and I land here
at s4, the fourth partial sum.

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And to get to s sub 4 to s sub 5, I'd be
adding

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a sub 5, but a sub 5 is even smaller than
a sub 4.

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So maybe that's somewhere over here.
What do I notice?

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Well, what I'm noticing here is that the
even partial

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sums, s sub 2 and s sub 4 are increasing.

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And the odd partial sums, s sub 1, s sub
3, s sub 5, they're decreasing.

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I can be a little bit more precise, right?

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This sequence, the sequence s sub 2 n
minus 1 is decreasing.

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This sequence, s sub 2 n, is increasing.
So these two sequences are monotone.

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The s sub evens

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is increasing, but bounded above by s sub
1.

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And the s of odds are decreasing, but
bounded below by, say, s sub 2.

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So I've got monotone and bounded
sequences.

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So by the monotone convergence theorem,
the sequence of even

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partial sums is increasing and bounded
above, and therefore the

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limit exists.

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And by the monotone conversion theorem the
sequence of odd partial sums is decreasing

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but bounded below, and consequently the
limit of s sub odd exists.

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Well, that's awesome, or not.

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What did I actually want to prove?

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Well, I actually wanted that the limit of
just the partial sums

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exists, I mean, I don't, I don't really
care about the sequence

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of even partial sums and the sequence of
odd partial sums.

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I just want to know what the sequence of
partial sums, does it have a limit?

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So how do I know that the odd and the even
sequences converge to the same thing?

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So what I know is that the limit of the
even

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partial sums exist and the limit of the
odd partial sums exists.

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But I just want to know about the limit of
the partial sums all together, right?

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The even terms are getting close to
something, the odd

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terms are getting close to something but

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are all the terms getting close to
something.

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Well to analyze this, I'm going to look
instead at

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this limit, I'm going to look at the limit
of

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the difference of the even and the odd
partial

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sums and there's two different ways to
calculate this limit.

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On the one hand

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[LAUGH]

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well I can just calculate what this term
is, this is the sum of the

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first 2 end terms, this is the sum of the
first 2n minus 1 terms.

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So if I add up the first 2n terms

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and subtract all but the last term that's
exactly

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the same then as just the 2 nth term,
which in this case is negative a sub 2n.

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So this limit is the same as this limit,
but what is

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this limit, right?

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Well I actually know what that one's equal
to because

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I assume that the limit of the nth term is
0.

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So

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[LAUGH]

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my, my assumption right the limit of
negative a sub 2n is

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the same as negative the limit of a sub n
which is 0.

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But now there's another way that I could
calculate this limit.

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This is the limit of a difference, which
is

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the difference of the limits provided the
limits exist.

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And in this case, they do, right?

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I'm assuming that these two limits exist.
So this is the limit of a difference.

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This is a difference of the limits.
But now,

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[LAUGH].

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I just calculated this a moment ago, to be
equal to 0, right?

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What I'm saying here is that the limit of
this

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sequence, and the limit of this sequence,
differ by 0.

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Which is to say that this limit equals
this limit, right?

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Which is just to say that the limit of s
sub n exists, because

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the limit of the even terms is equal to
the limit of the odd terms.

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So all of the terms together are getting
close

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to something in particular and that's what
I wanted.

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Right the limit of a partial sums exist
and that's

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exactly what it means to say that this
series converges

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right to say that a series converges is
just to

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say that the limit of the sequence of
partial sums exists.

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But wait, there's more.

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Let's think back to this picture, right?
This diagram

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of the number line with the partial sums
drawn on it.

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Well, the odd partial sums are decreasing,
and their limit is some value L.

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The even partial sums are increasing, and
their limit's the

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same thing, it's also L, because the
difference between even and

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odd partial sums are controlled by just
the terms in

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the sequence a sub k, and those terms have
limit 0.

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So the even partial sums are

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increasing to L, the odd partial sums are
decreasing to L.

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And what this means is that the value of
this series is between these two, right.

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Here's the value of the of the series.

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Here's the series, and all of the even
partial sums are below it

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and all of the odd partial sums are above
this true value, L.

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This is

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one of the best reasons to care about
alternating series.

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So not only is it really easy to

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determine if an alternating series
converges, the whole

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story just boils down to whether the limit
of a sub n is 0 or not.

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00:10:27,700 --> 00:10:31,580
But more than that, I can now give
explicit

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error bounds on the value of an
alternating series.

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If I compute the 2 nth and the 2n minus
1th partial sum, I know the true value

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of my series lands between these two
values.

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00:10:44,580 --> 00:10:47,770
And this makes alternating series great
for machines.

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When you're calculating a partial sum,
you're getting error bounds for free.

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[SOUND]