Series Geometrica

sChapter 31 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal

31 Geometric Series
Motivation
(I hope)

Geometric series are a basic artifact of algebra that everyone should know.1 I am teaching them here because they come up remarkably often with Markov chains. The finite geometric series formula is at the heart ofmany of the fundamental formulas of financial mathematics. All students of the mathematical sciences should be intimately familiar with this topic and have all the formulas memorized. following properties: A geometric series is a sum of either a finite or an infinite number of terms. Each term after the first term of a geometric series is a multiple of the previous term by some fixed constant, x.Geometric series can be characterized by the

Example Example

25 + 50 + 100 + 200 + 400 is a geometric series because each term is twice the previous term. 4 + 2 + 1 + .5 + .25 + .125 + .625 + ... is an (infinite) geometric series because each term is 1/2 the previous term.

Multiplication of a geometric series by a constant does not affect its nature. It is still a geometric series. Whetherit converges (actually adds up to anything) is unaffected. If x + x2 + x3 + x4 + ... = L, then CAx + CAx2 + CAx3 + CAx4 + ... = CAL.

1

Japanese children are thoroughly trained in geometric series before they enter pre1

school.

Chapter 31 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal

The Finite GeometricSeries
The most basic geometric series is 1 + x + x2 + x3 + x4 + ... + xn. This is the finite geometric series because it has exactly n + 1 terms. It has a simple formula:

Formula 1

The Finite Geometric Series

This formula is easy to prove: just multiply both sides by 1 - x. All but two terms on the left will cancel. It can be proven just as easily by induction (proving it is an exercisein Section 6).

Example

There is a simple fairy tale known to many people that I cannot tell here because this is a college text and it would be improper. However, if I were to tell it, it goes something like this: some ordinary bloke saves the king's life. The king, being at heart a regular guy, is grateful. He offers ordinary bloke a of his kingdom. But all ordinary bloke wants is that achessboard be brought and on its first square be a grain of wheat, and on the second square two grains of wheat; then four on the next square and so on. The king thinks this is nothing. He offers ordinary bloke one of his daughters to go with a of his kingdom. He offers both of his daughters (this is actually a very sneaky trick, but that is another fairy tale). However, all ordinary bloke wants is achessboard, and on its first square he wants a single grain of wheat. On the second square he wants 2 wheat grains. On the third square he wants 4 grains, and so on. The king tries to get him to go for something else, but ordinary bloke won't budge. Finally, the king says let it be done. The Chancellor of the Wheatery comes back and says there is not enough wheat! It turns out that the wheat wasall eaten by rats. For embarrassing the monarchy the king has ordinary bloke's head cut off with a rusty axe. The moral of this story is quite simple: if a King offers you one of his daughters, take her; you can always find some way of dumping her later. 2

Chapter 31 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. CargalHowever, we being serious academics can go ahead and ask how much wheat did that guy want anyway? Well, on the first square he wanted 1 = 20 grains. On the next square he wanted 2 = 21 grains. On the next square he wanted 4 = 22 grains, and so on. By the time he gets to the 64'th square he wants 263 grains which is over nine-quintillion grains. But that is just the last square; the total he wanted...