Consider sequences $\{a_n\}$ and $\{b_n\},$ where $a_{n}2$ as $n+$ and $b_h=(-1{)}^n.$ Determine which of the following sequences converge and which diverge. If a sequence converges, indicate ts limit$.$

$($a$)\{b_n\}$

$($c$)\{a_n+b_n\}$

$($c$)\{b_n^2\}$

$($b$)\{3a_n-1\}$

$($d$)\{\frac{1}{a^2+3}\}$

$($f$)\{\frac{\frac{h_0}{100}}{10}\}$