On-line Math 21

1.0.1  Series

There are many oddities involved in this infinite summation process, which we call an infinite series. An infinite series is an infinite sum. There are a few examples where we really can see the infinite sum, like the one mentioned in the section on Achilles and the tortoise.

Example 1 1+1/2+1/4+1/8+... = 2

You can see that this sum really does add up to 2 by marking a line segment of two ``units'' length

Example 2

1 1·2 + 1 2·3 + 1 3·4 +... = 1.

This series, called the telescoping series, is the easiest of all to see, once you know the trick:

1 n·(n+1) = 1 n - 1 n+1 .

So, you add them together:

1 1·2 + 1 2·3 + 1 3·4 +... = ( 1 1 - 1 2 )+( 1 2 - 1 3 )+( 1 3 - 1 4 )+( 1 4 - 1 5 )+... = 1 1 +(- 1 2 + 1 2 )+(- 1 3 + 1 3 )+(- 1 4 + 1 4 )+(- 1 5 +... )+... = 1.

1.0.2  Sequences

We need to get some terminology straight here. A series, or infinite series, is the infinite sum. The things you add up are the terms of the series. They form an infinite sequence. A sequence is just the listing, one number after another. A series is a summing of those numbers.

Mathematicians think of sequences as being easier to understand. Just one number after another. You can usually tell if the sequence is approaching something, like 1/n approaches 0, as n goes to ¥. Technically, a sequence is a function of the positive integers, but we write one as {a_n} . We say that a sequence {a_n} has a limit L as n® ¥, if the numbers (terms) approach L as n gets large.

Let's look at some examples. We usually really only care about limits of sequences, and whether they exist. For example, find the limits of the following sequences, if they exist:

Example 3 a_n = 1/2ⁿ

[Solution] 

Example 4 b_n = 2ⁿ

[Solution] 

Example 5 {c_n} = {3, 3.1, 3.14, 3.141, 3.1415, 3.14159,¼} (This should remind you of something).

[Solution] 

Copyright (c) 2000 by David L. Johnson.