When [tex]n=1[/tex], you have [tex]a_1=\dfrac3{2!}=\dfrac32[/tex].
When [tex]n=2[/tex], you have [tex]a_2=\dfrac{3^2}{4!}=\dfrac38[/tex].
Clearly, [tex]a_1>a_2[/tex].
Assume [tex]a_k<a_{k-1}[/tex]. Now when [tex]n=k+1[/tex], you have
[tex]a_{k+1}=\dfrac{3^{k+1}}{(2k+2)!}=\dfrac3{(2k+2)(2k+1)}\times\dfrac{3^k}{(2k)!}=\dfrac3{(2k+2)(2k+1)}a_k<a_k[/tex]
Therefore by induction the sequence is monotone (decreasing).
