1. ## Sequence Problem

Determine if the sequence is nondecreasing and if it's bounded from above.

$
a_n = \frac{{(2n + 3)!}}
{{(n + 1)!}}
$

So...

$
\begin{gathered}
a_n \leqslant a_{n + 1} \hfill \\
\frac{{(2n + 3)!}}
{{(n + 1)!}} < \frac{{(2(n + 1) + 3)!}}
{{((n + 1) + 1)!}} \hfill \\
\frac{{(2n + 3)!}}
{{(n + 1)!}} < \frac{{(2n + 5)!}}
{{(n + 2)!}} \hfill \\
\frac{{(n + 2)!}}
{{(n + 1)!}} < \frac{{(2n + 5)!}}
{{(2n + 3)!}} \hfill \\
\end{gathered}
$

Then I have no clue how the answer key got to this:

$
(2n + 5)(2n + 4) > n + 2
$

It also says that is not bounded because:

$
\frac{{(2n + 3)!}}
{{(n + 1)!}} = (2n + 3)(2n + 2)...(n + 2)
$

I don't quite follow what's going on there.

Thank you.

2. Hi,
Originally Posted by RedBarchetta
(...)
$
\frac{(n+2)!}{{(n + 1)!}} < \frac{{(2n + 5)!}}
{{(2n + 3)!}} \hfill \\
$

Then I have no clue how the answer key got to this:

$
(2n + 5)(2n + 4) > n + 2
$
Remember the definition of $n!$ : $n! = n\times(n-1)\times(n-2)\cdots 3\times 2\times 1$. Using this, one can simplify both sides of

$
\frac{(n+2)!}{{(n + 1)!}} < \frac{{(2n + 5)!}}
{{(2n + 3)!}} \hfill \\
$

LHS :

$\frac{(n+2)!}{{(n + 1)!}}=\frac{(n+2)\times{\color{blue}(n+1)\times n \times (n-1)\cdots \times 2 \times 1}}{{\color{blue}(n+1)\times n \times (n-1)\cdots \times 2 \times 1}}=n+2$

RHS :

$\frac{{(2n + 5)!}}
{{(2n + 3)!}} = \frac{(2n+5)\times(2n+4)\times{\color{blue}(2n+3)\ times(2n+2)\cdots2\times 1}}{{\color{blue}(2n+3)\times(2n+2)\cdots 2\times 1}}=(2n+5)(2n+4)$

It also says that is not bounded because:

$
\frac{{(2n + 3)!}}
{{(n + 1)!}} = (2n + 3)(2n + 2)...(n + 2)
$
Same idea as before : simplify.