www.mathcentre.ac.uk/.../The%20sum%20of%20an%20infinite%20series.mp4

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In this video, we'll look at
sums of infinite series.
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Will stop by revising what a
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finite series is. A finite
series is the sum of all the
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terms of a finite sequence.
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An example of a finite sequence
is something like this.
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Would have two K plus one
with K going from one to
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10.
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And that's equal to.
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What starts with three? Then
it goes 5 Seven, and
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so on. 21
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So an example of a finite series
is just all of these things
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added together, so that's 3 + 5
+ 7 plus all the way up to
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21. To do this we write Sigma,
which means the sum of and then
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the rule for the cake term. So
that's 2K plus one.
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And then we have to write from
where case starts from where it
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goes too. So that's from K
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equals 1. Up cake was 10.
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In this case, the sum of this
series is 120.
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An infinite series is
the sum of all the terms
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of an infinite sequence.
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So I'll give you an example of
an infinite sequence here we
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could have. One over 2 to
the KOK equals 1.
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To Infinity.
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And that is.
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A half a quarter.
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Knife.
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And so on. So the series
that goes with this sequence
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would be 1/2 + 1/4 plus
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an eighth. And so on.
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And again we can use Sigma
notation to write this.
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We put down Sigma
like we did before.
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Write down the rule for the
cake turn, so that's one
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over 2 to the K.
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And this time K starts at one
and goes on forever. So I put an
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Infinity on top of the Sigma.
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Now what could we mean by the
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sum of this series? Without
it, the first few terms and
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see what happens.
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So series
was half
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plus 1/4.
Put on a few more terms, this
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time 116th. One over 32.
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And so on.
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If I add up the first 2 terms.
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I'll get 3/4.
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If I add up the first three
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terms. I get
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the first four terms.
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I get 15 over 16.
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And I found out the first 5
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terms. I get 31 over 32.
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The sums of these first few
terms are called partial sums.
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We say the first partial sum is
just the first term on its own.
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The second partial sum is the
sum of the first 2 terms, so
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in this case 3/4 the third
partial sum is the sum of the
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first three terms. So in this
case 7/8 and so on.
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If we write down the partial
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sums. So that's
a half three
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quarters 7/8. 15 sixteenths
and so on.
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We can see that they formed
the beginning of an
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infinite sequence.
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The in terms of the sequence is
the NTH partial sum.
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And if you look at the sequence,
you can see that it gets closer
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and closer to one, so it has
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limit one. So that means as
you are done, more terms of
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the series.
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It gets closer and closer to
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one. So it makes sense to say
that this series has some one.
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So the series one over 2
KOK equals 1 to Infinity.
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Which is this?
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Equals 1.
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If we
have a
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general theories
like.
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AK.
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OK equals 1 to Infinity, so
that's just the Series A 1 + 82
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+ 83 plus a four and so on.
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We say that the series has a
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sum. If the NTH partial sums
converge to a limit.
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So that is the sequence
given by A1 A 1
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+ 82. Say 1 + 82 + 83. These
are the NTH partial sums, so we
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say that this.
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Series Has a sum. If this
sequence has a limit.
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Here's another infinite series
that has a sum.
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This time will have.
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The series one over K Times
K plus one with K going from
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one to Infinity.
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And that's equal to 1 over 1 *
2 first term.
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And then one over 2 * 3, then
one over 3 * 4.
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One over 4 * 5 and so on.
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Now we can split up each
individual term of this series.
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So we look at one term, one over
K Times K plus one.
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And I can break up the numerator
in 2K plus one minus K.
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So then I can break up the
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whole fraction. In 2K plus one
over K Times K plus one.
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That's minus K over K Times
K plus one.
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Now you can see the first bit of
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this. The whole numerator
cancels with some of the
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nominator so that K Plus one
cancels with this K plus one.
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And on this side that K cancels
with this K.
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So we get left with on this side
one over K.
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Now that side we get left with
minus one over K plus one.
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So now I can substitute this
back into our original series.
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To rewrite it.
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So the series one over K Times K
plus one from K equals 1 to
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Infinity can be rewritten like
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this. Well, for the first term,
cake was one.
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And we've just shown that this
can be written as one over K
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minus one over K plus one.
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So since K is one.
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For the first term, we get one
over one which is 1 - 1 over 2.
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So that's the first term. We do
the same for the second term,
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this time Kay is 2, so we get
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one over 2. Minus one over 2 +
1 which is 3.
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We carry on doing this so the
next one is 1/3 - 1/4.
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And so on.
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So from this we can workout the
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NTH partial sum. The sum of the
first end terms is 1 - 1/2
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+ 1/2 minus third.
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All the way up to the NTH term,
which is one over N minus one
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over N plus one.
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So this. Is the
sum of the first in terms.
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Now.
You can see that here a lot of
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things are counseling. You get
minus 1/2 + 1/2 - 1/3. Then you
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would get plus 1/3 and so on.
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So overall, you're left with 1 -
1 over N plus one.
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Which is equal to N plus
one over N plus one minus
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one over N plus one which
is just N over N plus one.
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So that was the end partial
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sum. So the sequence of partial
sums goes like this.
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It's the sequence and every one
plus one from N equals 1 to
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Infinity. And that goes one over
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2. Then 2/3
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3/4
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4/5 And so on.
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I can see that this sequence
gets closer and closer to one.
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In fact, the sequences
limit one.
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So the sequence of partial sums
for that last series has limit
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one. And that must mean.
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That this series.
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This series
has some
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one.
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Here's an example of a
series that doesn't have
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a son.
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Will have The series.
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Go from one to Infinity again.
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Of just one. So that's the
same as that just goes 1 +
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1 + 1 and so on forever.
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If we look at the sequence of
partial sums for this, it just
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goes or the first one is one,
then it's one plus one which is
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2. Then it's 1 + 1 + 1 which is
3, then four, and so on.
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Now, this sequence certainly
doesn't get closer to any
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number. In fact, it tends
to Infinity.
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If this happens, we say this
series doesn't have a sum.
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You might have spotted that
both the series we looked at
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got closer to 0.
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In fact, this is true always. If
a series has a son that the
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terms must get closer to 0.
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But this doesn't work the other
way round. Here's an example.
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Will have the series.
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One over K from K equals 1 to
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Infinity. So that's the series.
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1 + 1/2.
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Plus 1/3 as a quarter.
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And so on.
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Now, even though these terms get
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close to 0. The sequence of
partial sums for this tends to
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Infinity. The series, called the
harmonic series, and it doesn't
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have a sum. Now we'll have
a look at two very
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useful infinite series.
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First of all, we'll look at this
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one. It's one over K minus
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1 factorial. OK cause one to
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Infinity. So that's equal to one
over not factorial, which is
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one, so that's one.
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Then one over 1 factorial.
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Which again is one.
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Then one over 2 factorial which
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is 2. One over 3 factorial which
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is 6. And so on.
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to work at the end partial
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sums of this.
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I've done this already and the
end partial sums turn out to be,
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well. The first one obviously is
one, then two, then 2 1/2.
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And then you start needing
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the next one is.
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2.6 recovering so I'll put this
down to a few decimal places.
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The Next One is 2
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points. 708
Then three recurring.
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And the next one is.
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2.7176
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recovering. Will
do one more
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that's. Two points.
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718015
occurring.
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You can see that these numbers
are getting very close to
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something just below 3.
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In fact, these numbers keep
getting closer to E.
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So if you calculate.
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The series. For large number of
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terms. You can workout a very
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good approximation free. So this
makes the series very very
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useful. Here's
another useful
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infinite series.
This time we have.
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The sum of minus one to the K
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plus one. All times
for over 2K minus one.
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Again, K goes from one to
Infinity here.
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Let's workout the four over 2K
minus one, but first.
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When case one.
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This denominator becomes one.
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So this is 4 over 1 which is 4.
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Case 2. The nominator is 3,
so this becomes for over 3.
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The Next One is 4 over 5.
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We get for over 7.
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Forever 9 and so on.
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Now we just need to workout the
minus one to the K plus one bit.
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If K is odd, so that's for the
terms 135 and so on kaizad. Then
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keep this one is even.
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So minus 1 two and even number
is just one.
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So all these odd terms.
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A plus their positive.
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So put pluses in front of them.
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But if K is even.
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Than K Plus One is odd and minus
1 two an odd number is minus
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one. So all these even terms
have a minus before them.
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Again, we can we use our
calculators to workout with the
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NTH partial sums of the series
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On the end, partial sums turn
out to be the first one is easy,
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it's 4. The Next One is 2 and
2/3, so if I write the decimal,
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it's 2.6667 to a few decimal
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places. Then you really
start to need your
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Calculator. The next one is
3.4 than six recurring.
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The Next One is 2.895. Two
that will do goes on.
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The Next One is
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3.33. 97 that goes on as well.
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Then 2.97.
6046 and so on.
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And I'll do one more at 3.28.
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37
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38
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Now with this sequence it's a
bit hard spot which number is
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converging to. You could
probably spot it's converging to
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a number just above 3.
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In fact, it converges to π.
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So if you want to calculate π to
a large number of decimal
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places, all you have to do is to
workout a large and partial sum
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of this and that will give you a
good approximation for pie.
Title:
www.mathcentre.ac.uk/.../The%20sum%20of%20an%20infinite%20series.mp4
Video Language:
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 mathcentre edited English subtitles for www.mathcentre.ac.uk/.../The%20sum%20of%20an%20infinite%20series.mp4