Sequences

Introduction

A sequence can be thought of as a list of numbers written in a definite order.

$a_1, a_2, a_3, ... , a_n, ...$

It can also be defined as a function with the domain of the set of positive integers.

Limits of sequences

A sequence can have a limit $L$ which is written as:

$\displaystyle\lim_{n \to \infty} a_n$

If such a limit exists then the sequence converges. If the limit does not exist then the sequence diverges.

Limit laws and the squeeze theorem can also be applied here.

Increasing & decreasing

A sequence is said to be strictly increasing if:

$a_n \lt a_{n+1} \quad \forall n \ge 1$

A sequence is said to be strictly decreasing if:

$a_n \gt a_{n+1} \quad \forall n \ge 1$

A sequence is called monotonic if it is either strictly increasing or strictly decreasing.

Bounded sequences

A sequence is bounded above if there is a number $M$ such that:

$a_n \le M \quad \forall n \ge 1$

A sequence is bounded below if there is a number $M$ such that:

$M \le a_n \quad \forall n \ge 1$

If a sequence is bounded both above and below then it is called a bounded sequence.

Fun fact

Every bounded monotonic sequence is convergent.