Science:Infinite Series Module/Units/Unit 1/1.3 Infinite Series/1.3.01 Introduction to Infinite Series

Suppose that we are given an infinite sequence

${a_{n}}_{n = 1}^{\infty}$

and that we would like to add its elements together

$a_{1} + a_{2} + a_{3} + \dots + a_{n} + \dots$

which would produce an infinite series or an infinite sum, as was introduced in the previous lesson. Using sigma notation, we can write this sum as

$a_{1} + a_{2} + a_{3} + \dots + a_{n} + \dots = \sum_{k = 1}^{\infty} a_{k}$

This simple construction leads us to two immediate questions:

How do we know if the infinite sum produces a finite number?
If the given infinite sum does yield a finite number, what is it?

These two questions and their answers are the subject of this lesson and a key component of this module on sequences and infinite series.

Example

Consider the following example. Suppose we would like to determine the sum of the infinite series

$2 + \frac{1}{2} + \frac{1}{8} + \frac{1}{32} + \dots$

Indeed, we are faced with the two questions listed above: does this sequence yield a finite number, and if it does, what is it? Given a general problem, the answers to these two questions are not straightforward. In this lesson, we will introduce one approach to answer these two questions if the given sum has a special form. We will see that because the infinite sequence above has this particular form, we can calculate its sum quite easily.