Q$4).($a$)$ Using comparison test, check whether the following series are convergent or divergent.

$($i$)$

${\displaystyle \underset{n=0}{\overset{}{}}}\frac{3}{4^n+n}$

$(5$ marks$)$

$($ii$)$

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{n}{n^3+si{n}^2(n)}$

$($b$)$ Using alternating series test check whether the series

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{(-1{)}^{n+1}}{n}$

converges or diverges.

$(5$ marks$)$

$($c$)$ Use ratio test to show that the series

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{(-9{)}^n}{6^{2n+1}(n+1)}$

converges absolutely$.$

$(5$ marks$)$