Hello everyone. Welcome back. In this lecture we will be talking
about series and series representation. This will be something like
a little review for some people. So let's dive into it. If you're the objectives
are the following. We would like to recall infinite
series and their convergence. We would like examine geometric series and represent rational functions
as a geometric series. So let's start with the sequence first. The sequence is basically a list
of numbers in some definite order. If the limit of this sequence exists. To a as n gets larger and larger, then
we say that the sequence is convergent. So, let me use some examples a n
= n / n+1, as an increase you can see that these numbers are getting
closer and closer to the one. If you look at a n = 3 to the n. Which is basically 3, and then 9 and 27. It goes larger and larger,
it does not converge. In that case actually
we say it's divergent. It diverges to infinity. n equal to radical n, because 1,
radical 2, radical 3, radical n, it still increases. It may not that passed as n or
n squared, or three to the n, but it does increase to infinity so
it's also a divergent sequence. Like if I look at one over n squared,
it's going to be one, one over four, one over nine,
eventually it will decrease to zero. Now partial sums of a sequence
n are defined as follows. You look at first n
the terms in the sequence. We add them up. And that becomes our end partial sum. We call it sn. For example, s1 is the first a1. And then we look at the s2,
which is a1 + a2. s3, which is a1 + a2 + a3. So we keep adding. The terms from the sequence, and
then we make a new sequence. So we now have a partial sums, s1,
s2, s3, which is a new sequence. And here's the part that
serious coming to the play. If this partial sums a sense are actually
going closer and closer to some number, then we say the series, a1 + a2 +
a3 plus dot, dot, dot to infinity, in other words that infinite
sum is convergent and it is equal to that limit, s,
which is the limit of the partial sums. And we write sum notation from
cables from one to infinity, ak, it's the limit of a partial sum and
it's s. Now if. The partial sums do not converge. In other words, if the sequence s n
is broken then it was the series a k, k equals one from infinite sum and
up, it is a divergent series. So let m give this common example of
the Series one went to the case if you add them up, it becomes number one if
you add them one of the case requires If this is some of the number
chi square over six. If you -1 to the the k + 1 / k, which
is called alternating harmonic series, it because some number ln(2). But there are also some divergencies
occurs if you'll have this sequence, 3 to the k. If you keep adding them up,
you can think if it piles up and it doesn't converge to sum number. Sum of 3 to the k, sum of 2k + 1,
which is sum of odd numbers. Or even sum of 1/ks,
1 + 1/2 + 1/3 and so forth. As a sequence, one over k goes to zero,
but if you sum them up, they pile up and it becomes divergent. This last series here is
actually called harmonic series. Now, we say series is absolutely
convert it and if you take the absolute value of each perm in the sequence and
add them up and if it turns out this new series is convergent then the original
series is called absolutely convergent. Now one thing we have to mention
is that absolute convergence is actually a stronger convergence
implies convergence as well. There are a lot of convergence tests
out there where we use them to test a series of convergence, of divergent,
because finding partial sums and the limit of the partial sums is
not as efficient most of the time. So we use some tests to determine the convergence of a series, which I'm
not going to talk about in this lecture. There's one important series which
we're going to lose a lot in this week, is a geometric series. And let's start with
the geometric sequence first, geometric sequence is the following
you start with a number and you keep multiplying with some r,
so you start with a, next guy is a r, next guy is a r squared,
next guy is a r cubed, and so forth. If you add them up, if you add these
numbers up, infinitely many of them, you get this geometric series. And it turns out that geometric series is
sometimes convergent sometimes divergent. When is it convergent? It is convergent if it's multiplier
r absolute value less than one, have magnitude less than one. So let me give you an example. We talked about sum of one over two
to k it's right it's right here. Sum of one over two to k. The first guys 1 over 2,
then 1 over 4, 1 over 8. It is a geometric series. The starting number, a, is 1 over 2,
and the number you're multiplying, r, is actually half,
which has magnitude less than 1. Absolute value of r is less than 1. Which means this is going to be
a convergent geometric series, and you can use the formula a over 1-4, and it becomes 1 over 2 over 1 minus 1 over 2,
which is actually 1. So this whole sum sums
up to actually 1 series Now, series representation
is very important. What we do is here is a. If you have some rational function,
let's say you have 1 over 1- x. Can we express this rational function
as a series, as a geometric series? It turns out that, yes,
we can try to use our geometric series. So how do we do this? In this expression you can realize
that you can think of 1 as your a, and r as your x. Then 1 over 1- x is your a + AR, which is x, + AR squared,
which is x squared, and so forth. So we can express 1 over 1- x,
as infinite series but this equality will be only true for
some values of x because geometric series. In other words, this geometric
series will always work if that absolute value of r is x is less than 1. In other words 1/1-x Is
equal to this infinite sum, if and only if,
absolute value of x is less than 1. You can find series representation for rational functions in
the format of the following. You can 1 over sum 1
over x times 1- x over k. In other words,
we have some quadratic equation, quadratic polynomial denominator. What we can do with the following, we can
expand this using partial fractions and then each one,
they'll give us a geometric series and we connect this to geometric series
to get one series expansion. Now both of these expansions
must be convergent. In other words,
absolute value x has to be < 1, absolute value of x / 2 has to be < 1. In other words, absolute value of x
has to be < 1 is the common interval. As long as absolute value of x is < 1, this rational function can be expressed
as an infinite series right here. We can actually jump to the complex
functions if instead of actually have a z which is a complex number,
same expansion holds. In other words, a/1- z can be
written as a geometric series. But this geometric series will be
only convergent if Magnitude of z. This is an absolute value
[INAUDIBLE] is that the magnitude of the complex number is less than one. So the same idea works here as well. So what is we have
learned in this lecture? We have learned the definition
of the infinite series and when they're actually convergent. We have learned that Geometric series of
conversion if the multiplier has norm is another word for
magnitude in our case is less that 1. And you learn how to represent some
rational functions as a geometric series.