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Hello everyone.

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Welcome back.

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In this lecture we will be talking
about series and series representation.

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This will be something like
a little review for some people.

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So let's dive into it.

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If you're the objectives
are the following.

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We would like to recall infinite
series and their convergence.

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We would like examine geometric series and

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represent rational functions
as a geometric series.

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So let's start with the sequence first.

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The sequence is basically a list
of numbers in some definite order.

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If the limit of this sequence exists.

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To a as n gets larger and larger, then
we say that the sequence is convergent.

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So, let me use some examples a n
= n / n+1, as an increase you can

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see that these numbers are getting
closer and closer to the one.

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If you look at a n = 3 to the n.

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Which is basically 3, and then 9 and 27.

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It goes larger and larger,
it does not converge.

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In that case actually
we say it's divergent.

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It diverges to infinity.

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n equal to radical n, because 1,
radical 2, radical 3,

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radical n, it still increases.

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It may not that passed as n or
n squared, or three to the n, but

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it does increase to infinity so
it's also a divergent sequence.

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Like if I look at one over n squared,
it's going to be one, one over four,

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one over nine,
eventually it will decrease to zero.

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Now partial sums of a sequence
n are defined as follows.

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You look at first n
the terms in the sequence.

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We add them up.

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And that becomes our end partial sum.

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We call it sn.

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For example, s1 is the first a1.

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And then we look at the s2,
which is a1 + a2.

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s3, which is a1 + a2 + a3.

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So we keep adding.

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The terms from the sequence, and
then we make a new sequence.

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So we now have a partial sums, s1,
s2, s3, which is a new sequence.

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And here's the part that
serious coming to the play.

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If this partial sums a sense are actually
going closer and closer to some number,

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then we say the series, a1 + a2 +
a3 plus dot, dot, dot to infinity,

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in other words that infinite
sum is convergent and

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it is equal to that limit, s,
which is the limit of the partial sums.

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And we write sum notation from
cables from one to infinity,

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ak, it's the limit of a partial sum and
it's s.

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Now if.

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The partial sums do not converge.

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In other words, if the sequence s n
is broken then it was the series a k,

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k equals one from infinite sum and
up, it is a divergent series.

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So let m give this common example of
the Series one went to the case if you

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add them up, it becomes number one if
you add them one of the case requires

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If this is some of the number
chi square over six.

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If you -1 to the the k + 1 / k, which
is called alternating harmonic series,

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it because some number ln(2).

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But there are also some divergencies
occurs if you'll have this sequence,

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3 to the k.

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If you keep adding them up,
you can think if it piles up and

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it doesn't converge to sum number.

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Sum of 3 to the k, sum of 2k + 1,
which is sum of odd numbers.

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Or even sum of 1/ks,
1 + 1/2 + 1/3 and so forth.

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As a sequence, one over k goes to zero,
but if you sum them up,

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they pile up and it becomes divergent.

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This last series here is
actually called harmonic series.

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Now, we say series is absolutely
convert it and if you take the absolute

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value of each perm in the sequence and
add them up and if it turns out this

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new series is convergent then the original
series is called absolutely convergent.

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Now one thing we have to mention
is that absolute convergence

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is actually a stronger convergence
implies convergence as well.

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There are a lot of convergence tests
out there where we use them to test

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a series of convergence, of divergent,
because finding partial sums and

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the limit of the partial sums is
not as efficient most of the time.

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So we use some tests to determine

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the convergence of a series, which I'm
not going to talk about in this lecture.

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There's one important series which
we're going to lose a lot in this week,

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is a geometric series.

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And let's start with
the geometric sequence first,

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geometric sequence is the following
you start with a number and

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you keep multiplying with some r,
so you start with a,

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next guy is a r, next guy is a r squared,
next guy is a r cubed, and so forth.

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If you add them up, if you add these
numbers up, infinitely many of them,

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you get this geometric series.

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And it turns out that geometric series is
sometimes convergent sometimes divergent.

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When is it convergent?

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It is convergent if it's multiplier
r absolute value less than one,

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have magnitude less than one.

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So let me give you an example.

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We talked about sum of one over two
to k it's right it's right here.

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Sum of one over two to k.

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The first guys 1 over 2,
then 1 over 4, 1 over 8.

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It is a geometric series.

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The starting number, a, is 1 over 2,
and the number you're multiplying,

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r, is actually half,
which has magnitude less than 1.

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Absolute value of r is less than 1.

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Which means this is going to be
a convergent geometric series, and

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you can use the formula a over 1-4, and

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it becomes 1 over 2 over 1 minus 1 over 2,
which is actually 1.

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So this whole sum sums
up to actually 1 series

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Now, series representation
is very important.

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What we do is here is a.

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If you have some rational function,
let's say you have 1 over 1- x.

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Can we express this rational function
as a series, as a geometric series?

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It turns out that, yes,
we can try to use our geometric series.

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So how do we do this?

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In this expression you can realize
that you can think of 1 as your a,

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and r as your x.

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Then 1 over 1- x is your a + AR,

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which is x, + AR squared,
which is x squared, and so forth.

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So we can express 1 over 1- x,
as infinite series but

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this equality will be only true for
some values of x because geometric series.

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In other words, this geometric
series will always work if that

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absolute value of r is x is less than 1.

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In other words 1/1-x Is
equal to this infinite sum,

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if and only if,
absolute value of x is less than 1.

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You can find series representation for

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rational functions in
the format of the following.

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You can 1 over sum 1
over x times 1- x over k.

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In other words,
we have some quadratic equation,

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quadratic polynomial denominator.

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What we can do with the following, we can
expand this using partial fractions and

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then each one,
they'll give us a geometric series and

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we connect this to geometric series
to get one series expansion.

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Now both of these expansions
must be convergent.

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In other words,
absolute value x has to be < 1,

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absolute value of x / 2 has to be < 1.

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In other words, absolute value of x
has to be < 1 is the common interval.

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As long as absolute value of x is < 1,

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this rational function can be expressed
as an infinite series right here.

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We can actually jump to the complex
functions if instead of actually

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have a z which is a complex number,
same expansion holds.

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In other words, a/1- z can be
written as a geometric series.

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But this geometric series will be
only convergent if Magnitude of z.

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This is an absolute value
[INAUDIBLE] is that the magnitude

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of the complex number is less than one.

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So the same idea works here as well.

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So what is we have
learned in this lecture?

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We have learned the definition
of the infinite series and

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when they're actually convergent.

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We have learned that Geometric series of
conversion if the multiplier has norm

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is another word for
magnitude in our case is less that 1.

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And you learn how to represent some
rational functions as a geometric series.