Determine whether the sequence converges or diverges. If it converges, find the limit. If it diverges write NONE.
$a_n = \frac {(5n-1)!}{(5n+1)!}$
$\displaystyle\lim_{n \to \infty} a_n=?$
the answer is $0$, but I have no idea how to get the value.
$\endgroup$ 51 Answer
$\begingroup$We have
$$a_n=\frac{(5n-1)!}{(5n+1)!}$$
$$a_n=\frac{1.2.3...(5n-1)}{1.2.3...(5n-1)5n(5n+1)}$$
$$=\frac{1}{5n(5n+1)}$$
So,
$$\lim_{n\to+\infty}a_n=0.$$
$\endgroup$
