Problem $3$:

$($a$)(5$ points$)$ Find the limit: ${\displaystyle \underset{k=1}{\overset{k=}{}}}\frac{1}{k(k+1)}.$

$($b$)(8$ points$)$ Determine if the series converges: ${\displaystyle \underset{k=1}{\overset{k=}{}}}(\sqrt[\phantom{2}]{k+1}-\sqrt[\phantom{2}]{k}{)}^2.$

$($c$)8$ points$)$ Determine if the series converges: ${\displaystyle \underset{k=2}{\overset{k=}{}}}\frac{1}{(lnk{)}^k}.$

$($d$)$ points$)$ Determine if the series converges: ${\displaystyle \underset{k=2}{\overset{k=}{}}}\frac{lnk}{1+3k^{1\mathrm{.}5}}.$

$($e$)(8$ points$)$ Find the value of $0\mathrm{.}123456$dots, i$.$e$.$ the limit of ${\displaystyle \underset{k=1}{\overset{k=}{}}}\frac{k}{{10}^k}.$