Test the series for convergence $or$ divergence using the Alternating Series Test.

${\displaystyle \underset{n=0}{\overset{}{}}}\frac{sin((n+\frac{1}{2}))}{3+\sqrt[\phantom{2}]{n}}$

Identify $b_n.$

Evaluate the following $limit.$

$\underset{n}{lim}{b}_n$

Since $\underset{n}{lim}{b}_n,0$ and $b_{n+1},b_n$ for all $n,$