Sequence

analysis

Definition

Sequence

A sequence is an arrangement of elements $a_0,a_1,a_2,\dots$, which can alternatively be written as:

$$(a_n{)}_{n\ge 0}$$

Sequences can be interpreted as functions $a:\mathbb{N}\to \mathbb{R}$ with $a(n)=a_n$.

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

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

$$(a_n{)}_{n\ge k}$$

with $k$ being the starting index.

Almost All

A statement holds for almost all $n\in \mathbb{N}$ if it is valid for all but finitely many exceptions.

Convergence

Convergent

Definition

Convergent Sequence

A sequence $(a_n{)}_{n\ge 0}$ is called convergent if it has a limit.

Example: The limit of the below sequence is:

$$\underset{n\to \infty }{\lim }{a}_n=a$$

which can also be written as:

$$a_n⟶a$$

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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 $(a_n{)}_{n\ge 0}$ is called monotonically decreasing if

$$\forall n\in \mathbb{N}:a_{n+1}\le a_n$$

A sequence $(a_n{)}_{n\ge 0}$ is called strictly monotonically decreasing if

$$\forall n\in \mathbb{N}:a_{n+1}<a_n$$

Monotonically Increasing

Monotonically Increasing

A sequence $(a_n{)}_{n\ge 0}$ is called monotonically increasing if

$$\forall n\in \mathbb{N}:a_{n+1}\ge a_n$$

A sequence $(a_n{)}_{n\ge 0}$ is called strictly monotonically increasing if

$$\forall n\in \mathbb{N}:a_{n+1}>a_n$$

Bound

Upper Bound

Definition

Upper Bound

A sequence $(a_n{)}_{n{\ge }_0}$ is called bounded above if there exists a real number $S$ such that:

$$\exists S\in \mathbb{R}:\forall n\in \mathbb{N}:a_n\le S$$

Any such number $S$ 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 $(a_n{)}_{n{\ge }_0}$ is called bounded below if there exists a real number $I$ such that:

$$\exists I\in \mathbb{R}:\forall n\in \mathbb{N}:a_n\ge I$$

Any such number $I$ called an lower bound of that sequence. If the sequence is not bounded below, $\inf a_n=-\infty$.

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

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