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

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Suppose that we are given an infinite sequence

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

and that we would like to add its elements together

${\displaystyle a_{1}+a_{2}+a_{3}+\ldots +a_{n}+\ldots }$

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

${\displaystyle a_{1}+a_{2}+a_{3}+\ldots +a_{n}+\ldots =\sum _{k=1}^{\infty }a_{k}}$

This simple construction leads us to two immediate questions:

1. How do we know if the infinite sum produces a finite number?
2. 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

${\displaystyle 2+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}+\ldots }$

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.