Better Living Through Mathematics! with Professor Polly Ethylene
Infinite series is one of the most fascinating subjects in mathematics.
But before we talk about infinite series, I would like to ask a question about infinity.
What do you get when you add an infinite number of things?
A.V. Geekman, do you have an answer up that sleeve of yours?
Well, the more things you add up, the bigger the answer gets.
So if you were to add up an infinite number of things
you would always get an infinitely large answer.
Are you sure about that A.V.?
Well, no matter what size the things are
an infinite number of them will always be infinity.
For example, an infinite number of "ones" is infinity times one, or infinity.
An infinite number of one-halves is infinity times one-half.
One half of infinity is still infinity.
Even an infinite number of one-billionths is one-billionth of infinity
which is still infinity!
Ah ha!
That's extremely perceptive of you A.V.
It is true that an infinite number of anything, no matter how small, is still infinitely large.
So, another question to you.
Is it possible to add an infinite number of things and get a finite number?
Well, based upon our previous discussion, I would say not.
Well then, you would be wrong!
Oh really?
Now A.V., as you pointed out
if you add an infinite number of anything, no matter how small
the sum is still infinite.
But what if the things you add get progressively smaller?
I ... don't know.
All right then!
Let's add an infinite number of things, where each thing is one-half of the previous thing.
Let's say, for example, that you walk halfway to the wall.
Then you walk half of the remaining distance
then half of that distance
and half of that distance, and so on.
No matter how many times you keep doing that, you will still never quite reach the wall.
So it IS possible to add an infinite number of things and get a finite number!
Now you're cooking with gas!
I'll give you one more example.
Let's take the fraction nine-tenths
which you can write as the decimal number zero point nine.
Now let's add nine-hundredths
and nine-thousandths
and so on.
As you can see, we can keep doing this as long as you like
but you will never get a number bigger than one.
We say the "limit" of this series of additions is one.
So, let's talk about what a series is.
A series is just a list of things added together.
These things can be numbers
or they can be expressions or formulas which create numbers.
Let's call these cute little things that are added together "terms".
So a series is just a sum of terms.
Now, if I wanted you to add a bunch of terms for me
I might write down all of the terms in a list.
That list of terms is called a "sequence"
and when you add all the terms in a sequence together, it is called a "series".
Now, what if the sequence has a very large number of terms?
In fact, what if the sequence is infinite?
It might be easier to come up with a formula for creating the terms
instead of having to list each one.
Now wouldn't that be a good idea?
Wonderful!
Now, being the good perceptive professor that you are
you might have noticed there was a pattern to the terms I wrote on your list.
If we number the terms one, two, three, etc. then each term is just twice that number.
So we can specify this sequence by writing a formula.
This formula says that each term
which we will call "a" numbered with a little subscript "n"
which tells us which term in the sequence it is
is just two times n.
So for instance, the twentieth term, "a" sub twenty
is just two times twenty, or forty.
Now isn't that easier than listing every term?
Of course, we could make the formula for creating terms as complicated as we like
such as
or
or even something really complicated!
But the point is, if we can come up with a formula for creating the terms in the sequence
we can write the sequence in a very compact form and save a lot of paper.
Thank you for showing the class how to write an infinite sequence, Professor Ethylene
but weren't you going to explain to the class about infinite series?
Well thank you Professor Von Schmohawk for reminding me of that fact.
As I mentioned, a "series" is just the terms of a "sequence" added together.
Now, there is a nice way to write a series
using what we mathematicians call "summation notation".
Here is the summation notation for the first five terms in the sequence
which I wrote down for the good Professor Von Schmohawk.
The "summation symbol" is the capital Greek letter "sigma".
It indicates that the term to the right
is added over and over again
each time, using a different value of n.
In this case, n starts at one.
This value of n is then used to calculate the first term of the series.
Since n is one 2n is equal to two times one
or two.
So two is the first term in the series.
Then we increase n by one, and do it again.
This calculates the second term of the series.
We keep doing this until n finally reaches the value at the top of the summation symbol.
So this summation notation is another way of writing these five terms added together.
The sum of the terms is thirty
so this finite series is equal to thirty.
What if instead of stopping when n equals five, we went on forever?
In this case, instead of the five at the top, we would put a little infinity sign.
This would then be an "infinite" series.
Now we would keep adding terms forever.
In this series the terms get bigger and bigger, so the sum is obviously infinite.
But even if the terms were all the same number, the sum would still be infinite.
Take for example an infinite series where all the terms are the number one
or one-half.
In fact, adding any number that's not zero an infinite number of times gives you infinity.
So as long as the terms grow or stay the same
an infinite number of them will always sum up to infinity.
But what would happen if each term was smaller than the previous term?
Let's take an infinite series of the terms one over two to the nth power.
In this series the first term is one over two to the first power, or one-half.
The second term is one over two squared, or one-fourth.
The third term is one over two cubed, or one-eighth,
and so on.
Let's draw a picture of what happens when we add the terms in this series.
Start by drawing a square with a length and height of one
so that the square has an area of one.
Now, the first term of our series is one-half
so draw a rectangle with an area of half of the square
and place it in the square.
Now the second term of the series is one-fourth.
Let's draw a square with an area of half of the rectangle
and place it in the square.
The third term of our series is one-eighth
so let's draw a rectangle with an area of half of the previous square
and place it in the square.
This process can be repeated forever without overflowing the square.
As the little squares and rectangles continue to add up
their total area becomes closer and closer to the area of the big square.
The combined area of the terms gets closer and closer to one.
If you could add an infinite number of these terms, the total area would be exactly one.
So we say that this series "converges" to one.
In other words, this series is "convergent".
Convergent series are very useful.
Some numbers like pi can only be calculated by using convergent infinite series.
Here are the first few terms of an infinite series which can be used to calculate pi.
Of course we can't actually add an infinite number of terms
unless we had an infinite amount of time.
However, we can make our answer as accurate as we like by simply adding enough terms.
But will all series converge as long as each term is smaller than the previous one?
Well, let's try a series with the terms one over n.
Now, the first term in this series is one divided by one
or one.
The second term is one divided by two, or one-half.
The third term is one-third, and so on.
Each term is smaller than the previous term.
But it turns out that this series does NOT converge.
Even though the terms get smaller and smaller, they will still add up to infinity.
We say that this series "diverges".
Perhaps it seems strange that some series with decreasing terms converge to a number
while other series with decreasing terms diverge to infinity.
It is not always obvious which series will converge or diverge.
Let's take a closer look at this series to see why it never converges.
Let's write down the first few terms of this infinite series.
Now, let's make little stacks equal in height to each term in the series.
Notice that the first term of the series is equal to one-half plus one-half.
The second term in the series is also one-half.
Now notice that the next two terms, one-third and one-fourth
are each at least as big as one-fourth.
So, if we add them together, their sum will be at least as big as one-fourth plus one-fourth
or one-half.
Now the next four terms, one-fifth, one-sixth, one-seventh, and one-eighth
are each at least as big as one-eighth.
So, if we add them together, their sum will be at least as big as four times one-eighth
or, once again, one-half.
Likewise, the next eight terms, one-ninth, one-tenth, one-eleventh, one-twelfth, one-thirteenth
one-fourteenth, one-fifteenth, and one-sixteenth
are each at least as big as one-sixteenth.
So when we add them together, their sum will be bigger than eight times one-sixteenth
or, once again, one-half.
Likewise, the sum of the next sixteen terms is bigger than one-half
and the sum of the next thirty-two terms is bigger than one-half
and so on.
We can keep going on forever, grouping the terms into sums which equal more than one-half.
So the sum of this infinite series is at least as big as
the sum of an infinite number of one-halves
which is of course, infinite.
This particular series is called a "harmonic series"
because its terms are similar to the harmonics of a musical note.
Oh, I diverge!
Although the harmonic series is interesting
it is not very useful because its sum never converges.
Are there any questions?
Hulk Moosemasher, what is your question?
Professor Ethylene, are infinite series useful?
Why yes Hulk, infinite series are very useful!
There are many things which can only be calculated by using infinite series.
For example, the ratio of the circumference of a circle to its diameter is pi.
For many centuries, people measured circles but could never determine exactly
what this ratio was to an accuracy of more than a few decimal places.
But with the help of an infinite series
we can determine pi to any degree of accuracy we like.
The more terms in the series we add, the more accurate our answer gets.
Infinite series are also used to calculate trigonometric functions such as sine and cosine
which are very useful in determining the angles and lengths of triangles
as well as exponential functions, logarithms
and many other mathematical functions which are used in engineering, science, and math.
I hope this answered your questions.
And remember, just like a ninety degree angle
I'm always RIGHT!