Limit Inferior/Examples
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Examples of Limits Inferior
Sequence of Reciprocals
Let $\sequence {a_n}$ be the sequence defined as:
- $\forall n \in \N_{>0}: a_n = \dfrac 1 n$
The limit inferior of $\sequence {a_n}$ is given by:
- $\ds \map {\liminf_{n \mathop \to \infty} } {a_n} = 0$
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 inferior of $\sequence {a_n}$ is given by:
- $\ds \map {\liminf_{n \mathop \to \infty} } {a_n} = -1$
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 inferior of $\sequence {a_n}$ is given by:
- $\ds \map {\liminf_{n \mathop \to \infty} } {a_n} = 0$
Limit Inferior 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 inferior of $\sequence {a_n}$ is given by:
- $\ds \map {\liminf_{n \mathop \to \infty} } {a_n} = -1$
This is not the same as:
- $\ds \inf_{n \mathop \ge 1} {\paren {-1}^n \paren {1 + \dfrac 1 n} }$