Chapter 9.1, Problem 3QCE

Consider sequence ${\text{{a}}_{\text{n}}\text{}}$ and ${\text{{b}}_{\text{n}}\text{}}$ , where $a_n\to 2$ as $n\to +\infty$ and $b_n={\left(-1\right)}^n$ . determine which of the following sequence converge and which diverge. If a sequence converges, indicate its limit.

$\begin{array}{l}\left(\text{a}\right)\text{}\left\{b_n\right\}\\ \left(\text{b}\right)\{3a_n-1\}\\ \left(\text{c}\right)\left\{b_n^2\right\}\\ \left(\text{d}\right)\{a_n+b_n\}\\ \left(\text{e}\right)\left\{\frac{1}{a_n^2+3}\right\}\\ \left(\text{f}\right)\left\{\frac{b_n}{1000}\right\}\end{array}$