A sequence is any function who's domain is an interval of integers. Sequences are useful because they allow for modelling discrete structures easily.

Notation

Ellipsis

Sequences can informally be written using an ellipses for example is a sequence of infinite terms. The symbol "" is called an ellipsis. It is shorthand for "and so forth".

If the sequence ends at a certain value it can be written using the ellipsis notation with the final term or two after the ellipsis for example .

Explicit Formulas

Sequences can be written explicitly as a formula depending on their index, for example the previous sequence could be defined as follows:

Where is the position in the sequence.

Specifying an Interval

When stating the interval, for an explicit formula we can use the following syntax:

Where can be any arbitrary equation, and represents the "upper bound" of the interval and where is the index variable and is the lower bound, can be any variable and the lower bound can be any integer .

Example

This represents the sequence and or and .

Implicit Formulas

Sequences can also be written in terms of previous terms, for example the Fibonacci sequence can be written using an implicit formula as follows:

Notice to define this implicit formula we need to defined distinct values called the "base cases" for and .

Terminology

Similarly to sets sequences have elements called terms, these terms are ordered by their index in the sequence, for example the sequence defined earlier as , has the terms . And given the previous set the term at index is or , this can alternatively be written as .

Vergence

Every sequence either converges or diverges

• The sequence converges if is finite, where is the term of the sequence.
• The sequence diverges if it does not converge.

Types of Divergence

The sequence is said to diverge to if and is said to diverge to if , where is the term of the sequence.

Sequences of Functions

If a sequence is defined in terms of a function for example , if converges to some constant , then the sequence necessarily converges. Conversely though if converges that doesn't mean that converges, this is because sequences operate only on integers where as functions can operate on any number.

For example given the function and the sequence , the sequence converges to where as the function always oscillates between and .

Relation to a Series

For any infinite sequence (that is a sequence with an upper bound of ) it can be written as a series, for example given the sequence it's corresponding series is written as .