Q$3).($a$)$ Use divergence test to show that the series

${\displaystyle \underset{n=0}{\overset{}{}}}\frac{4n^{200}+n^{\text{'}}}{10+2n^{\text{'}}}$

diverges for any lixed integer $t,$ where $t>200.$

$(4$ marks$)$

$($b$)$ Consider the geometric series written in the form

${\displaystyle \underset{n=0}{\overset{}{}}}a{r}^n$

$($i$)$ Derive conditions under which the series converges and give the value the series converges to$.$

$(5$ marks$)$

$($ii$)$ Hence, show that the series

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{4^{n+1}}{9^{n-2}}$

converges and find the value of the series.

$(5$ marks$)$

$($c$)$ Check whether the telescopic series

${\displaystyle \underset{n=0}{\overset{}{}}}\frac{1}{3n^2+12n+9}$

converges or divereges.

$(6$ marks$)$