$Q2.$

$(a)$ Determine whether the following series converges $or$ diverges. $Ifitis$ convergent, evaluate $itif$

possible $todoso$ with the techniques we've learned. And $ifitis$ not possible $to$ evaluate, but $is$

still convergent, explain why you cannot evaluate $it.$

$\frac{4}{21}+\frac{4}{45}+\frac{4}{77}+$cdots$+\frac{4}{4n^2+20n+21}+$cdots

$(b)$ Consider the following statement:

Let $\{a_n{\}}_n=1^{}be$ a sequence $of$ real numbers. Suppose there $is$ another

sequence $\{b_n{\}}_n=1^{}$ such that $a_n=b_n-b_{n+2}$ for all $n1.If{\displaystyle \underset{n=1}{\overset{}{}}}{a}_{n}is$

convergent then $\underset{n}{lim}{b}_n$ exists.

$Is$ this statement true? $Ifso,$ prove $it(a$ suggestion would $beto$ mimic proofs $of$ any similar

results from our lesson$\frac{s}{e}$xamples$).If$ instead $itis$ false, state a clear counterexample: $a$

sequence $\{a_n{\}}_n=1^{}$ such that ${\displaystyle \underset{n=1}{\overset{}{}}}{a}_{n}is$ converges but $\underset{n}{lim}{b}_n$ doesn't exist.