I need to prove that the limit of the sequence is as shown(0):
1.$\lim_{n \to \infty} nq^n =0 ,|q|<1 $
2.$\lim_{n \to \infty}\frac{2^n}{n!}$
but I need to do this using the convergence tests. With the second sequence I tried the "ratio test", and I got the result
$\lim_{n \to \infty} \frac{2}{n+1} $
which means that L in the ratio test is 0 and so it proves that the sequence converges, but how now should i prove that the limit is indeed 0? I can't use the L'Hopital's rule.
and for the first sequences I am not sure where to start.
can you help please?