We're given a sequence defined by the recursive relation: $$a_n = \sqrt{a_{n-1}a_{n-2}}$$
$a_1$ and $a_2$ are positive constants.
We have to show the following:
The sequences $\{ b_n \} = \{ a_{2n-1} \}$ and $\{ c_n \} = \{ a_{2n} \} $ are monotonic, and if one is increasing, the other is decr...9
Let $ \ \mathbb{N} = \{ 0,1,2,3,4,...\} \, $, $ \ O = \{ n \in \mathbb{N} : n \text{ is odd} \} \ $ and $ \ T: O \to O \ $ be such that, for all $ \ n \in \mathbb{N} \, $,
\begin{align*}
T(8n+1) & = 6n+1 \\
T(8n+3) & = 12n+5 \\
T(8n+5) & = 2n+1 \\
T(8n+7) & = 12n+11
\end{align*}
For each $ \ n \i...
In mathematics, a topological space
X
{\displaystyle X}
is said to be limit point compact or weakly countably compact if every infinite subset of
X
{\displaystyle X}
has a limit point in
X
.
{\displaystyle X.}
This property generalizes a property of compact spaces. In a metric space, limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces, however, these three notions of compactness are not equivalent...
Sometimes I come across some mathematicians' CVs, in which they report Paul Erdős number. I was wondering whether in mathematics communities this number is taken seriously, and to what extent?
Ref: Erdős Number
