Sequence

analysis

Definition

Sequence

A sequence is an arrangement of elements , which can alternatively be written as:

Sequences can be interpreted as functions with .

The numbers that make up the sequence are called the terms or elements of the sequence, and is called the index of the sequence term.

If needed, one can also specify from which index the counting starts, e.g.:

with being the starting index.

Convergence

Convergent

Definition

Convergent Sequence

A sequence is called convergent if it has a limit.

Example: The limit of the below sequence is:

which can also be written as:

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Divergent

Definition

Divergent Sequence

A sequence without limits is called divergent.

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Monotonicity

Monotonically Decreasing

Monotonically Decreasing

A sequence is called monotonically decreasing if

A sequence is called strictly monotonically decreasing if

Monotonically Increasing

Monotonically Increasing

A sequence is called monotonically increasing if

A sequence is called strictly monotonically increasing if

Bound

Upper Bound

Definition

Upper Bound

A sequence is called bounded above if there exists a real number such that:

Any such number is called an upper bound of that sequence.

The smallest upper bound is called the supremum of that sequence.

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Lower Bound

Definition

Lower Bound

A sequence is called bounded below if there exists a real number such that:

Any such number called an lower bound of that sequence. If the sequence is not bounded below, .

The greatest upper bound is called infimum the of that sequence.

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