Limit Superior/Inferior of Sequences of Real Numbers Review
Limit Superior/Inferior of Sequences of Real Numbers Review
We will now review some of the recent material regarding the limit superior and limit inferior of sequences of real numbers.
- On The Limit Superior and Limit Inferior of a Sequence of Real Numbers page we said that if $(a_n)_{n=1}^{\infty}$ is a sequence of real numbers then the Limit Superior of $(a_n)_{n=1}^{\infty}$ is defined as:
\begin{align} \quad \limsup_{n \to \infty} a_n = \lim_{n \to \infty} \left ( \sup_{k \geq n} \{ a_k \} \right ) \end{align}
- Similarly, the Limit Inferior of $(a_n)_{n=1}^{\infty}$ is defined as:
\begin{align} \quad \liminf_{n \to \infty} a_n = \lim_{n \to \infty} \left ( \inf_{k \geq n} \{ a_k \} \right ) \end{align}
- On the Alternative Definitions for the Limit Superior/Inferior of a Sequence of Real Numbers we saw another useful alternative definition for the limit superior and limit inferior of a sequence $(a_n)_{n=1}^{\infty}$ which were:
\begin{align} \quad \limsup_{n \to \infty} a_n = \inf_{n \geq 1} \left \{ \sup_{ k\geq n} \{ a_k \} \right \} \end{align}
(4)
\begin{align} \quad \liminf_{n \to \infty} a_n = \sup_{n \geq 1} \left \{ \inf_{k \geq n} \{ a_k \} \right \} \end{align}
- On The Connection Between the Limit Superior/Inferior of a Sequence of Real Numbers page we established a very nice relationship between the limit superior and limit inferior of a sequence of real numbers. We saw that:
\begin{align} \quad \limsup_{n \to \infty} (-a_n) = -\liminf_{n \to \infty} a_n \end{align}
- We then looked at two comparison theorems on the Comparison Theorems for the Limit Superior/Inferior of Sequences of Real Numbers page. First, for any sequence of real numbers $(a_n)_{n=1}^{\infty}$ we have that:
\begin{align} \quad \liminf_{n \to \infty} a_n \leq \limsup_{n \to \infty} a_n \end{align}
- Secondly, if $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ are both sequences such that $a_n \leq b_n$ for all $n \in \mathbb{N}$ (in fact, for sufficiently large $n$ works) then we have that:
\begin{align} \quad \liminf_{n \to \infty} a_n \leq \liminf_{n \to \infty} b_n \quad \mathrm{and} \quad \limsup_{n \to \infty} a_n \leq \limsup_{n \to \infty} b_n \end{align}
- On the Properties of the Limit Superior/Inferior of a Sequence of Real Numbers we looked at some nice properties of the limit superior and inferior of a sequence of real numbers. If $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ are sequences of real numbers then:
\begin{align} \quad \limsup_{n \to \infty} (a_n + b_n) \leq \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_n \end{align}
(9)
\begin{align} \quad \liminf_{n \to \infty} (a_n + b_n) \geq \liminf_{n \to \infty} a_n + \liminf_{n \to \infty} b_n \end{align}
- Furthermore, if $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ are positive sequences of real numbers, then we can further conclude that:
\begin{align} \quad \limsup_{n \to \infty} (a_nb_n) \leq \left ( \limsup_{n \to \infty} a_n \right ) \left ( \limsup_{n \to \infty} b_n \right ) \end{align}
(11)
\begin{align} \quad \liminf_{n \to \infty} (a_nb_n) \geq \left ( \liminf_{n \to \infty} a_n \right ) \left ( \liminf_{n \to \infty} b_n \right ) \end{align}
- On the The Limit Superior/Inferior of the Ratio of Terms in Positive Sequences of Real Numbers we saw that if $(a_n)_{n=1}^{\infty}$ was a positive sequence of real numbers then:
\begin{align} \quad \liminf_{n \to \infty} \frac{a_{n+1}}{a_n} \leq \liminf_{n \to \infty} (a_n)^{1/n} \leq \limsup_{n \to \infty} (a_n)^{1/n} \leq \limsup_{n \to \infty} \frac{a_{n+1}}{a_n} \end{align}
- We then looked at an extremely value theorem on the Limit Superior/Inferior Convergence Criterion for Sequences of Real Numbers page. We noted that for a sequence of real numbers $(a_n)_{n=1}^{\infty}$ that:
\begin{align} \quad (a_n)_{n=1}^{\infty} \: \mathrm{converges \: to \:} A \in \mathbb{R} \quad \Leftrightarrow \quad \liminf_{n \to \infty} a_n = A = \limsup_{n \to \infty} a_n \end{align}
- Similarly, on the Limit Superior/Inferior Proper Divergence Criterion for Sequences of Real Numbers we saw that:
\begin{align} \quad (a_n)_{n=1}^{\infty} \: \mathrm{properly \: diverges \: to \:} \infty \quad \Leftrightarrow \quad \liminf_{n \to \infty} a_n = \infty = \limsup_{n \to \infty} a_n = \infty \end{align}
(15)
\begin{align} \quad (a_n)_{n=1}^{\infty} \: \mathrm{properly \: diverges \: to \:} -\infty \quad \Leftrightarrow \quad \liminf_{n \to \infty} a_n = -\infty = \limsup_{n \to \infty} a_n = \infty \end{align}