Assumed knowledge

The content of the modules:

• Algebra review
• Functions I.

Motivation

We encounter sequences at the very beginning of our mathematical experience. The list of even numbers

\[ 2,\ 4,\ 6,\ 8,\ 10,\ \dots \]

and the list of odd numbers

\[ 1,\ 3,\ 5,\ 7,\ 9,\ \dots \]

are examples. We can `predict' what the 20th term of each sequence will be just by using common sense.

Another sequence of great historical interest is the Fibonacci sequence

\[ 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ 55,\ 89,\ 144,\ \dots \]

in which each term is the sum of the two preceding terms; for example, \(55 = 21 + 34\). In this case it is somewhat more difficult to predict the 20th term, without listing all the previous ones.

Sequences arise in many areas of mathematics, including finance. For example, we can invest $1000 at an interest rate of 5% per annum, compounded annually, and list the sequence consisting of the value of the investment each year:

\[ \$1000,\ \$1050,\ \$1102.50,\ \$1157.63,\ \$1215.51,\ \dots \] (rounded to the nearest cent).

Sequences can be either finite or infinite. For example,

\[ 2,\ 4,\ 6,\ 8,\ 10 \]

is a finite sequence with five terms, whereas

\[ 2,\ 4,\ 6,\ 8,\ 10,\ \dots \]

continues without bound and is an infinite sequence. We usually use \(\ldots\) to denote that the sequence continues without bound.

For a given infinite sequence, we can ask the questions:

• Can we find a formula for the general term of the sequence?
• Does the sequence have a limit, that is, do the numbers in the sequence get as close as we like to some number?

For example, we can see intuitively that the terms in the infinite sequence

\[ 1,\ \dfrac{1}{2},\ \dfrac{1}{3},\ \dfrac{1}{4},\ \dfrac{1}{5},\ \dots \]

whose general term is \(\dfrac{1}{n}\), are approaching 0 as \(n\) becomes very large.

A finite series arises when we add the terms of a finite sequence. For example,

\[ 2 + 4 + 6 + 8 + \dots + 20 \]

is the series formed from the sequence \(2,4,6,8,\dots,20\).

An infinite series is the `formal sum' of the terms of an infinite sequence. For example,

\[ 1 + 3 + 5 + 7 + 9 + \dotsb \]

is the series formed from the sequence of odd numbers. We can spot an interesting pattern in this series. The sum of the first two terms is 4, the sum of the first three terms is 9, and the sum of the first four terms is 16. So we guess that, in general, the sum of the first \(n\) terms is \(n^2\).

For a given infinite series, we can ask the questions:

• Can we find a formula for the sum of the first \(n\) terms of the series?
• Does the series have a limit, that is, if we add the first \(n\) terms of the series, does this sum get as close as we like to some number as \(n\) becomes larger?

If it exists, this limit is often referred to as the limiting sum of the infinite series. In this module, we examine limiting sums for one special but commonly occurring type of series, known as a geometric series.

Sequences and series are very important in mathematics and also have many useful applications, in areas such as finance, physics and statistics.

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