For a series $S,$ let $S=1-\frac{1}{9}+\frac{1}{2}-\frac{1}{25}+\frac{1}{4}-\frac{1}{49}+\frac{1}{8}-\frac{1}{81}+\frac{1}{16}-\frac{1}{121}+$cdots$+a_n{a}_n=\{\begin{array}{ccc}\frac{1}{2^{\frac{n-1}{2}}}& ifnis\text{o}\text{d}\text{d}& \\ \frac{-1}{(n+1{)}^2}& ifnis\text{e}\text{v}\text{e}\text{n}& \end{array}$

Which of the following statements are true?

I. $S$ converges because the terms of $S$ alternate and $\underset{n}{lim}{a}_n=0.$

II$.S$ diverges because it is not true that $|a_{n+1}|<mi>|</mi><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>|</mi>$ for all $n.$

III. $S$ conrverges although it is not true that $|a_{n+1}|<mi>|</mi><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>|</mi>$ for all $n.$

$($A$)$ None

$($B$)$ I only

$($C$)$ II only

$($D$)$ III onlv