Professor Dave here, let’s converge a little.
Earlier is this course, we introduced the concept of a sequence as a particular list
of numbers, and we looked at some basic types, like arithmetic sequences, geometric sequences,
and even some more interesting ones like the Fibonacci sequence.
We saw how these can be expressed in a few different ways, as beyond simply listing the
terms, they can be abbreviated using a particular expression or formula.
We also expanded on this to look at series and summation notation.
If you’ve never seen these things before, go back and check out this tutorial now, because
we are going to be building on that knowledge over the next few tutorials, since we now
understand calculus.
First, we want to be able to assess whether a sequence is convergent or divergent, so
let’s find out what that means.
Take the sequence A sub N equals N. That means that the first term is one, then two, and
so forth through the integers.
We want to find out what this sequence does in the limit of N approaching infinity.
For this one it’s easy to see that the sequence approaches infinity as N approaches infinity,
so we would call this sequence divergent, because the limit does not exist.
Then let’s look at something like A sub N equals N over N plus one.
This will give us one half, two thirds, three fourths, and so forth.
We can pretty quickly see that this sequence will approach one but never quite get there,
so the limit of this sequence as N approaches infinity, is one.
Because the limit exists, as a finite number, we would call this sequence convergent.
There are some sequences for which it’s a little trickier to determine the limit,
and with these it may be useful to apply some of the limit laws we already know, as they
could help us manipulate things into a form where the limit is easier to assess.
Sometimes we can use special techniques, like using L’Hospital’s rule on the related
function, or something called the squeeze theorem.
This one says that if there are three sequences, A, B, and C, and B is always between A and
C, then if A and C both have a limit of L, then B must also have a limit of L, because
it is being squeezed in between these ones.
Some sequences will be divergent without going to infinity, like negative one to the N power.
This just alternates between positive and negative one forever, so there is no limit.
So once again, we look at what a sequence does as N approaches infinity.
If the sequence asymptotically approaches some finite number, it is convergent.
If it does not do that, it is divergent.
Now that we understand convergence and divergence for sequences, let’s apply this understanding
to series.
First we have to recall what a series is.
When we try to add up all the terms in an infinite sequence, we get an infinite series.
This can be represented by listing out all the terms, but it can also be abbreviated
using the summation notation we already learned, with the upper case sigma.
We can talk about convergent and divergent series just like we did for sequences.
Starting again with a simple example, simply N, we can easily see that if we were to add
up all the positive integers, which themselves go to infinity, certainly their sum will also
be infinite.
This can be demonstrated by forming a new sequence, where the first term is the first
term of the series, and then the sum of the first two terms of the series, and then the
first three terms, and four terms, and so forth.
The limit of this sequence is the sum of the original series, and the sequence will go
to infinity, so the original series is clearly divergent.
Can any infinite series be convergent?
That is, can an infinite number of terms add up to give a finite number?
Well when we looked at improper integrals, we saw exactly this sort of thing happening,
with a limit of integration going to infinity but still yielding a finite value for the
area under the curve, so we shouldn’t be too surprised that convergent infinite series exist.
Take for example one over two to the N. Listing the first few terms we get one half, one fourth,
one eighth, and so on, approaching zero.
Let’s use the same technique as before, creating a new sequence and listing the first
term of the series, then the sum of the first two, which is three fourths, then the sum
of the first three, which is seven eighths, and if we continue on in this manner, we see
that this sequence approaches one.
Therefore the sum of this series is equal to one, and we can say that it is convergent.
Different types of series will have different requirements for convergence.
Take this generalized geometric series.
If we recall the definition of a geometric sequence, it is one in which you multiply
each term by some constant to get the next one, so starting with A, we get AR, then ARR,
or AR squared, then AR cubed, and so forth.
So a geometric series is just the sum of these terms, or the sum of A times R to the (N minus one).
We will find that if the absolute value of R is greater than one, a geometric series
will be divergent, because these terms will get bigger and bigger, and therefore go to infinity.
If equal to one, we are just adding up infinitely many A terms, which will also be divergent.
But going back to the sequence, if the absolute value of R is less than one, as it is raised
to bigger and bigger exponents, each term will get smaller, and the limit of the sequence
will be zero, so the series, or the sum of the terms, will be convergent, and the sum
will be equal to A over the quantity (one minus R).
In fact, an important theorem states that in order for a series to be convergent, the
sequence that generates the terms in the series must have a limit of zero.
This makes sense, because if the limit were any other number, or if the sequence itself
was divergent, there would be no way for the terms in the series to add up to a finite value.
And even if the limit of the sequence is zero, it is still not a guarantee that the related
series will converge.
So sometimes, we can look at a series and state that it is divergent simply by showing
that the sequence it is derived from does not converge to zero.
Take something like N squared over the quantity (five N squared plus four) from one to infinity.
To assess the series, let’s just look at the sequence of numbers we get from this expression,
without considering their sum.
What is the limit of this sequence as N approaches infinity?
Well we need some algebraic manipulation first.
Let’s divide both top and bottom by N squared so that we aren’t plugging in so many infinities.
On top we get one, and on the bottom we get five plus four over N squared.
Now we plug in infinity and this term goes to zero, leaving us with one fifth.
The sequence does not converge to zero, so the sum of this series that includes all the
terms from that sequence must be divergent.
Now that we understand the concepts of convergence and divergence as they apply to both sequences
and series, we are ready to move forward and learn more about series.