$($d$)$ By finding the limit$,$ show that the sequence

$a_n=\{\frac{5n^{997}-6n^{998}+4^{999}}{8n-n^{999}+2}{\}}_n=0^{}$

converges.

$(3$ marks$)$

$($e$)$ Use ratio test to show that series

${\displaystyle \underset{n=0}{\overset{}{}}}\frac{n!}{5^n}$

diverges.

$(4$ marks$)$

$($f$)$ Using Monotone Convergence Theorem $($MCT$)$ check whether the sequence

$a_n=\{\frac{1}{n^2+{}^{}}{\}}_n=1^{}$

converges. If it converges, what is the limit of this sequence?

$(4$ marks$)$

$($g$)$ Use the Squeeze Theorem for Sequences to show that the sequence

$a_n=\{\frac{2^n}{n!}{\}}_n=0^{}$

converges.

$(4$ marks$)$