Show the given sequence is eventually strictly increasing or

[hank]

eventually strictly decreasing.

I'm not sure that I understand how to handle factorials in this problem:

{n!/(3^n)} for n=1 to +inf

Can someone show me how to deal with them?

 

[pka]

Decreasing??
\(\displaystyle \L\begin{array}{l}
\frac{{a_{n + 1} }}{{a_n }} = \frac{{\frac{{\left( {n + 1} \right)!}}{{3^{n + 1} }}}}{{\frac{{n!}}{{3^n }}}} = \frac{{n + 1}}{3} \\
n \ge 3\quad \Rightarrow \quad \frac{{a_{n + 1} }}{{a_n }} > 1\quad \Rightarrow \quad a_{n + 1} > a_n \\
\end{array}\)

 

[hank]

I have to show if it is eventually strictly increasing or if it is eventually strictly decreasing.

How do you eliminate the factorial?

 

[pka]

Where are your algebra skills?

\(\displaystyle \L \frac{{\frac{{\left( {n + 1} \right)!}}{{3^{n + 1} }}}}{{\frac{{n!}}{{3^n }}}} = \left[ {\frac{{\left( {n + 1} \right)!}}{{3^{n + 1} }}} \right]\left[ {\frac{{3^n }}{{n!}}} \right] = \left[ {\frac{{\left( {n + 1} \right)\left( n \right)!}}{{n!}}} \right]\left[ {\frac{{3^n }}{{3^{n + 1} }}} \right] = \frac{{n + 1}}{3}.\)

 

[hank]

Cool, thanks.

I don't recall every doing anything with factorials like that in algebra.

 

[stapel]

I don't recall every doing anything with factorials like that in algebra.

They're probably gonna be important for the remainder of this topic, and possibly in later sections of your book, so you might want to read up on them.

Factorials are generally pretty easy to work with, once you get the hang of 'em.

Eliz.