Question Determine the convergence or divergence of the series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{2}{\sqrt[\phantom{2}]{n}+0\mathrm{.}1}.$ Justify your answer.

Solution Let $a_n=\frac{2}{\sqrt[\phantom{2}]{n}+0\mathrm{.}1}$ and $b_n=\frac{1}{\sqrt[\phantom{2}]{n}}.$ Note $b_{n}0.$

$L=\underset{n}{lim}\frac{a_n}{b_n}=\underset{n}{lim}$

Note $L$

By the $p-$test $($with $p=$

So$,{\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{\sqrt[\phantom{2}]{n}+0\mathrm{.}1}$ as well, by the limit comparison test.