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  • 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