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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