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Every sport has its standouts. Some notables include professional Chessboxer Andy 'The Rock' Costello, toe wresting champion Paul 'Predatoe' Beech, and korfball all star Amy Turner. Math has its all star standouts as well. The arithmetic and geometric sequences were a couple of notable standouts in the last chapter. Just like sequences, arithmetic and geometric series tend to dominate the landscape.
The arithmetic series is one of the simplest series we can come up with. In terms of common restaurant menu items, the arithmetic series is a burger. Although it many be spiced up with bacon, feta cheese, and some type of questionable special sauce, it appears on every restaurant menu in one form or another. We need to understand this series type backward and forward.
An arithmetic series is a series whose terms form an arithmetic sequence. Simple enough, right? If we wrote down an arithmetic sequence, we could replace all of the commas with plus signs to get an arithmetic series.
1 + 3 + 5 + 7 + ...
is an arithmetic series because
1, 3, 5, 7,...
gets from one number to the next by adding 2 and only by adding 2. Because we are adding the same amount every time, this is an arithmetic series.
1 + 3 + 6 + 10 + ...
is not an arithmetic series because
1, 3, 6, 10, ...
gets from one number first by adding 2, then by adding 3, then by adding 4, etc. Because we're adding different amounts, this isn't an arithmetic series.
We have some good news. Just like a plain, old burger, we already know everything there is to know about the convergence of arithmetic series.
To start with, a constant series is an arithmetic series where the difference between successive terms is d = 0. The constant series
only converges if a = 0. That's a pretty bold statement. Take a look at a couple problems to understand why.
If arithmetic series are the burgers of restaurant menus, then geometric series are the hotdogs. They aren't sold in every restaurant, but we all reminisce about our 6-year-old days when we bite into a diner dog.
It would be neat if a geometric series was made from adding up circles, pentagons, and Pac-Man shaped polygons called nomnomagons. But they're much simpler. A geometric series is a series whose terms form a geometric sequence.
is a geometric series because
is a geometric sequence. To get from one term to the next in a geometric sequence you must multiply by the same number (called the common ratio) each time; in this example the common ratio is ½.
is not a geometric series because its terms do not comprise a geometric sequence.
There are two magic formulas we need to know, and know really well, for dealing with geometric series:
A finite geometric series is the same thing as a partial sum of an infinite geometric series. This means, if we can sum finite geometric series, we'll be able to find partial sums of infinite geometric series.
It doesn't matter where the first term of a sequence begins. Much the same it doesn't matter too much where the first term of a geometric series begins. For the sake of making Sigma notation tidy and the math as simple as possible, we usually assume a geometric series starts at term 0. Mathematicians are some of the laziest people around. They just solve math problems day after day.
The first 1 term of the series corresponds to i = 0.
ar0 + ar1 + ar2 + ...
The first 2 terms of the series correspond to i = 0 and i = 1:
ar0 + ar1 + ar2 + ...
The first 3 terms of the series correspond to i = 0, i = 1, and i = 2:
ar0 + ar1 + ar2 + ...
Continuing in this manner, the first n terms of the series correspond to the terms from i = 0 to i = (n – 1).
ar0 + ar1 + ar2 + ... + arn – 1 + arn + ...
This means the nth partial sum of the geometric series
Sn = ar0 + ar1 + ar2 + ... + arn – 1
This is the sum of n terms. The index ranges from 0 to n – 1.
Be Careful: When finding the nth partial sum of a geometric series, the index ranges from 0 to (n – 1).
The index does not get up to n.
We promised a magic formula for finite geometric series. This time, we are going to pull a lemming out of an empty reusable grocery bag. Don't worry. The magic formula works as promised.
The nth partial sum of the geometric series
is given by
Be Careful: n is the number of terms, not the highest exponent.
There's a good chance you'll never need to know where this formula comes from. It's more important that you know how to use the formula properly.
The nth partial sum of a series is the sum of the first n terms of that series. Think of the finite geometric series
a + ar + ar2 + ... + arn – 1
as the first n terms of an infinite geometric series, so
Sn = a + ar + ar2 + ... + arn – 1.
We have no idea who came up with this idea or how they came up with it, but if we multiply both sides of this equation by (1 – r) and rearrange a little we end up with the magic formula.
Sn = a + ar + ar2 + ... + arn – 2 + arn – 1
(1 – r)Sn = (1 – r)(a + ar + ar2 + ... + arn – 2 + arn – 1).
If we expand the right-hand side and distribute the r, a LOT of things cancel.
Putting this back together with the left-hand side of the equation,
(1 – r)Sn = a(1 – rn).
Dividing both sides by (1 – r) gives us the magic lemming formula:
If you don't feel satisfied or enlightened right now, we understand. At least the formula comes from somewhere.
Be Careful: When entering expressions like this into your calculator, make sure you get all the parentheses in the correct places. Parenthetical errors are banes of existence for both mathematicians and computer programmers.
In order to use our magic lemming formula for finite geometric series, we need to know r, a and n. The 'ran' parameters of a geometric series are simple to find, as long as we remember what they are.
Keeping these in mind, we won't be left 'ran'ning in circles.
Geometric series are the hotdogs of the series world. But hotdogs can be prepared in an number of ways: with or without relish or chili; with or with a bun; boiled, baked or grilled... Likewise, we can write the same finite geometric series several different ways:
(I) a + ar + ar2 + ... + arn – 1
Although these expressions all give the value of the same geometric series, practically speaking (I) and (II) are completely different from (III).
The difference is pretty easy to see. Suppose someone gives you a and r and asks you to evaluate the series when n = 10000. With expression (III) you can pull out your calculator and do it. With expressions (I) and (II) you'd be more likely to have a temper tantrum and throw the calculator in a public water fountain.
We say the expression
is in closed form.
We liked closed form expressions in math. This means evaluating the formula always takes the same amount of work no matter how big n gets.
Whether n is 10, 1000, or 10000000000000, we
Closed form expressions save us all time, frustration and gray hair.
a + ar + ar2 + ... + arn – 1
aren't in closed form. In order to evaluate these suckers, we have to
That's unpleasant, to say the least. More realistically, it's like sorting a truckload of green and yellow grains of sand by grain color.
As much as we love sigma notation, the closed form is often more useful.
Imagine the infinite ways you can top a hotdog. Relish looks kind of bland after you consider what strawberry sauce and sprinkles could do to a hotdog.
Infinite geometric series, which are just specially dressed finite geometric series, have some wacky properties that can make them interesting. The standard infinite geometric series looks like
We already know for a geometric series the nth partial sum is
where r is the ratio of consecutive terms, a is the first term, and n is the number of terms.
Use the formula for Sn to find
(a) |r| < 1
(b) 1 < |r|
(c) r = 1 (hint: look at the expanded geometric series)
(d) r = -1 (hint: look at the expanded geometric series)
(a) Using the formula for Sn,
As n approaches ∞, since |r| < 1 the quantity rn approaches 0.
(b) Again we use the formula for Sn. Since |r| > 1, the quantity rn approaches ∞ as n approaches ∞.
The numerator of the fraction will get farther from 0 without bound as n approaches ∞, so this limit doesn't exist.
(c) Following the hint, look at the expanded series. When r = 1 the geometric series looks like
Since every power of 1 is 1, this is the constant series
a + a + a + ...
The nth partial sum is Sn = a · n.
does not exist. We found this result earlier when looking at a constant series
(d) Looking at the expanded series, when r = -1 we get
which is the alternating series
a – a + a – a + a – a.
The partial sums are
S1 = a = a
S2 = a – a = 0
S3 = a – a + a = a
S4 = a – a + a – a = 0
and so on.
Since the partial sums bounce back and forth between two values,
does not exist.
The last example covered all cases for r. We weren't playing the role of Oedipus here, leading you blindly down a dangerous path. We know that the sum of an infinite series, if it exists, is the limit of the partial sums Sn. In the previous exercises we found that
only exists when |r| < 1, in which case
This means when |r| < 1, the sum of the infinite geometric series
In all other cases (when |r| ≥ 1) the limit of partial sums
doesn't exist. This means when |r| ≥ 1 the sum of the infinite geometric series
doesn't exist either.
All of this means that an infinite geometric series converges when the ratio r has magnitude strictly less than 1. An infinite geometric series diverges in all other cases.
Finding the sum of an infinite geometric series is easier than finding a partial sum, because we only need to know a and r. We don't need to worry about how many terms there are. There are infinitely many.
The sum of a convergent geometric series is
where a is the first term of the series and r is the ratio.
We know all we need to know about geometric series. As a nifty bonus, we can use geometric series to better understand infinite repeating decimals. This is something you can rub in your former math teachers' faces.
Does 0.99999... equal 1?
First we have to figure out what 0.99999... means. If we break it up by individual decimal places,
0.99999... = 0.9 + 0.09 + 0.009 + 0.0009 + 0 .00009 + ....
Convert each term to a fraction. Now
This is an infinite geometric series with and . The sum of the series is
We've just proved that
.99999... = 1.
Haha! Take that 8th-grade math teacher that took off 5 points from your test when you wrote '1' instead of '0.9999...'.
The same trick can be used to turn other infinite repeating decimals back into rational numbers.