Hints :
(i) Decompose:
$\displaystyle (\sin 3n)(\cos n)=\dfrac{1}{2}(\sin 4n+\sin 2n)$
(ii) Use the Dirichlet test.
Fernando Revilla
Hints :
$\displaystyle \sin \dfrac{1}{\sqrt{n}}\sim \dfrac{1}{\sqrt{n}}\;(n\rightarrow +\infty)$
$\displaystyle \log \dfrac{n+1}{n-1}=\log\left(1+\dfrac{2}{n-1}\right)\sim \dfrac{2}{n-1}\; (n\rightarrow +\infty)$
P.S. Perhaps, it should be better to open new threads for new problems. Thanks.
Fernando Revilla