Subtitles section Play video Print subtitles 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