I'm having trouble with the sequence: $\displaystyle a_n=\frac{1*3*5*...*(2n-1)}{n!}$
It seems that the limit must exists since each factor$\displaystyle a_n=\frac{1}{1}*\frac{3}{2}*\frac{5}{3}*\frac{2n-1}{n}$ has a finite limit.
It is easy to see that is...
$\displaystyle a_{n+1} = a_{n} \frac{2n+1}{n+1}$ (1)
... so that is...
$\displaystyle \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}} = 2$ (2)
... and that means that is...
$\displaystyle \lim_{n \rightarrow \infty} a_{n} = \infty$ (3)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$