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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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    7/8. If I add up
    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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    that had sums had terms they
    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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    You can use your Calculator
    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
    to use your Calculator so
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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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    are. I've done this already.
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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
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