For each of the following pairs of functions f$($n$)$ and g$($n$),$ decide whether we have f$($n$)$ in $\backslash$Theta $($g$($n$)),$ or f$($n$)$ in O$($g$($n$)),$ or f$($n$)$ in $\backslash$Omega $($g$($n$)).$ If f$($n$)$ in $\backslash$Theta $($g$($n$)),$ you will not earn any point by selecting f$($n$)$ in O$($g$($n$))$ or f$($n$)$ in $\backslash$Omega $($g$($n$)),$ since f$($n$)$ in $\backslash$Theta $($g$($n$))$ tells more than each of the other two. $($a$)$ g$($n$)=$

i$=1$

n

$$

i$,$f$($n$)=$n

$2$

$\backslash$times $($lnn$)$

$.($b$)$ g$($n$)=1\mathrm{.}001$

n

$,$f$($n$)=$n

$100$

$.($c$)$ g$($n$)=2\backslash$times n$!,$f$($n$)=(2\backslash$times n$)!.($d$)$ f$($n$)=$

i$=1$

$2$n

$$

i

$2$

$,$g$($n$)=$

i$=1$

n

$$

$(2$i$)$

$2$

$.$