Determine if the sequence $\{a_n\}$ converges or diverges. Find the limit if the sequence converges.

$a_n=(-\frac{1}{3}{)}^n+\frac{1}{\sqrt[\phantom{2}]{5^n}}$

Select the correct choice below and fill in any answer boxes within your choice.

A$.$ The sequence $\{a_n\}$ converges. The limit is $\underset{n}{lim}{a}_n=.($Simplify your answer.$)$

B$.$ The sequence $\{a_n\}$ diverges.