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.