# Science:Infinite Series Module/Units/Unit 1/1.1 Infinite Sequences/1.1.01 Introduction to Infinite Sequences

An infinite sequence of numbers is an ordered list of numbers. Examples could include

1. the positive integers: 1, 2, 3, 4, ...
2. the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, ...
3. a sequence defined by Newton's Method:
${\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}},\ x_{0}\ \mathrm {given} }$

For the purposes of the ISM, we will only consider sequences with real numbers.

## Notation

We often will use the notation

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

to denote a general infinite sequence, whose terms are a1, a2, a3, ... . For example, in the case of the Fibonacci sequence,

{\displaystyle {\begin{aligned}a_{1}&=1\\a_{2}&=1\\a_{3}&=2\\a_{4}&=3\\\end{aligned}}}

## Key Questions for This Lesson

Given a sequence of numbers, ${\displaystyle \{a_{n}\}_{n=1}^{\infty }}$, we could ask the following questions:

• How do we know if the infinite sequence converges to 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.

## Examples

If we consider examples 1 and 2 above, then we can see that by inspection, the sequences does not converge to a finite number because successive terms in the sequences are increasing.

However in some cases, it can be more difficult to establish whether the sequence converges. Depending on the form of ${\displaystyle f(x)}$ in Example 3 above (Newton's Method), it can be difficult or impossible for us to determine whether this sequence converges by inspection. Indeed, we need a more rigorous method to establish convergence, which we will explore next.