Question $4$

$($a$)$ State the following.

$($i$)$ Limit comparison test.

$($ii$)$ Test for divergence.

$2,2$

$($b$)$ Prove that a positive term series either converges or diverges.

$($c$)$ Use the alternating series test to test for the convergence of a series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{(-1{)}^n}{2n!}.$

$($d$)$ Use the ratio test to determine the convergence of the series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{n^2+2n+1}{3^n+2}.$