Limit Superior/Examples

Examples of Limits Superior

Sequence of Reciprocals

Let $\sequence {a_n}$ be the sequence defined as:

$\forall n \in \N_{>0}: a_n = \dfrac 1 n$

The limit superior of $\sequence {a_n}$ is given by:

$\ds \map {\limsup_{n \mathop \to \infty} } {a_n} = 0$

Divergent Sequence $\paren {-1}^n$

Let $\sequence {a_n}$ be the sequence defined as:

$\forall n \in \N_{>0}: a_n = \paren {-1}^n$

The limit superior of $\sequence {a_n}$ is given by:

$\ds \map {\limsup_{n \mathop \to \infty} } {a_n} = 1$

Farey Sequence

Consider the Farey sequence:

$\sequence {a_n} = \dfrac 1 2, \dfrac 1 3, \dfrac 2 3, \dfrac 1 4, \dfrac 2 4, \dfrac 3 4, \dfrac 1 5, \dfrac 2 5, \dfrac 3 5, \dfrac 4 5, \dfrac 1 6, \ldots$

The limit superior of $\sequence {a_n}$ is given by:

$\ds \map {\limsup_{n \mathop \to \infty} } {a_n} = 1$

Limit Superior of $\paren {-1}^n \paren {1 + \dfrac 1 n}$

Let $\sequence {a_n}$ be the sequence defined as:

$\forall n \in \N_{>0}: a_n = \paren {-1}^n \paren {1 + \dfrac 1 n}$

The limit superior of $\sequence {a_n}$ is given by:

$\ds \map {\limsup_{n \mathop \to \infty} } {a_n} = 1$

This is not the same as:

$\ds \sup_{n \mathop \ge 1} \paren {-1}^n \paren {1 + \dfrac 1 n}$