Let $a_n=\frac{2n}{5n+1}.$

$(a)$ Determine whether $\{a_n\}is$ convergent $or$ divergent. $Ifitis$ convergent, find the $limit.(If$ the quantity diverges,

enter DIVERGES.$)$

$(b)$ Determine whether ${\displaystyle \underset{n=1}{\overset{}{}}}{a}_{n}is$ convergent.

Converges; the $limitof$ the terms $a_{n}is$ a constant $asn$ goes $to.$

Converges; the series $is$ a constant multiple $of$ a geometric series

Diverges; the sequence $a_n$ diverges $asn$ goes $to.$

Diverges; the $limitof$ the terms $a_{n}is$ not $0asn$ goes $to.$

Diverges; the series $is$ a constant multiple $of$ the harmonic series.