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PROFESSOR: Hi, welcome
back to recitation.

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Today we are going to talk about
some, an infinite series

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and discuss it's convergence.

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So in particular I have
this infinite series.

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The sum from n equals 1 to
infinity of 1 divided by the

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product n times n plus 1.

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So what I'd like you to do is to
compute a few terms of the

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series, compute a few partial
sums, and use that to get a

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sense for what you think
the series is doing.

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Is it converging?

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Is it diverging?

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If it's converging, can you
figure out what value it's

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converging to?

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So why don't you pause the
video, take some time to try

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that out, see what you
get, come back and

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we can do it together.

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So this is a nice series.

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It has terms that are
easy to compute.

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And I've taken the liberty of
computing a few of them in

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advance, and I've put
them up over here.

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So for n from 1 to 5, the terms
that we're adding up are

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1 over n times n plus 1.

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So that's when n equals
1, that's 1 over 1

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times 2, which is 1/2.

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When it is 2, it's 1 over
2 times 3, which is 1/6.

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Then we've got 1/12, 1/20,
1/30, and so on.

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So those are the things
we're adding up.

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And then the partial sums, the
nth partial sums. Well, the

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first one is just the first
term, which is 1/2.

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The second one, we take the
first term and the second term

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and we add them together.

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So 1/2 plus 1/6 is 2/3.

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The third one, we take the first
three terms and add them

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together and that
gives us 3/4.

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And OK, so I computed the
first five partial

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sums here as well.

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Now, if you look at this column,
so remember that the

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limit, that the value of an
infinite series is defined to

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be the limit of its partial
sums. So if we want to know

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what is the value of this
infinite series that we

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started with, does it converge,
does it diverge,

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what we have to do to figure
that out is we have to take

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its partial sums and we have
to compute their limit.

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And if we, if their
limit doesn't

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exist, then it diverges.

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If their limit does exist, then
the sum of the series is

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equal to what that value
of that limit is.

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And if you look at these terms
here, you'll see that they,

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there's a little bit of
a pattern here, right?

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So these, this is 1/2,
2/3, 3/4, 4/5, 5/6.

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That's a pretty nice sequence
of numbers.

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It's, you know, we could
probably guess at this point

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that the next one is going to
be 6/7 then 7/8 and so on.

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So that would be a guess.

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One way we can actually
prove this is, so we

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have this guess that--

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let me write it down.

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Guess is that Sn is equal
to n over n plus 1.

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Now if you wanted to confirm
this guess, what you'd have to

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do is you have to just figure
out how could you prove that.

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Well, one thing you can do is
you can say, well, Sn plus 1

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is equal to Sn plus the
next term, right?

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So in our case, Sn plus 1 is
equal to Sn plus the next

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term, the n plus first term,
which in our case is 1 over n

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plus 1 times n plus 2.

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So, all right, so that's not
maybe obvious what to do with

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this, but you could split this
up, really you can split it up

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by partial fractions.

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And you can write this as say Sn
plus, so if you split this

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up by partial fractions what
you'll get is that it's

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exactly equal to 1 over n plus
1 minus 1 over n plus 2.

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And from here it's easy to see
that, well, if Sn is equal to

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n over n plus 1, then this will
be equal to 1 minus 1

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over n plus 2, which is n
plus 1 over n plus 2.

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And so using the process known
as mathematical induction, you

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have that it follows for
all values of n.

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So because of this nice
expression for the term, it's

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easy to see that this pattern
will continue forever.

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OK, and so that, you know,
that's just a sketch of how

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you would prove this.

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Now once you've proven this,
it's easy to see, that the,

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once you know that this is true,
it's easy to see what

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this limit is right?

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As n goes to infinity, this
just approaches 1 and that

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means the series converges and
the limit of the series is 1.

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So here we have a nice example
of a series that converges and

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where it actually is possible
to compute the

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limit of this series.

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This isn't possible for most,
for all series or even for

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most series.

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Even ones with fairly nice
terms, it's often very

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difficult to figure out what
their limit is, but in this

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case it's not hard to do and
we have precisely that the

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value of the series is 1.

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So I'll end there.

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