$1.)$ #$1-$For the following two sequences determine whether it converges or diverges, and if there is convergence then find. the limit $($give reasons for all conclusions$)$:

i$.)a_n=\frac{(-1{)}^{n+1}n}{n+\sqrt[\phantom{2}]{n}}.$

ii$.)a_n=2^{-n}cosn.$