Answer
Function satisfies conditions for integral test
Series diverges
Work Step by Step
In order integral test can be applied $\sum ^{\infty }_{n=1}a_{n}$
$f(x) $ must be positive ,continuous and decreasing for $x \geq 1$
We see that $f\left( x\right) =\dfrac {2}{3x+5}$ is positive and continuous for $x \geq1$ and in order to find if the function is decreasing we need to find derivative of function
$f'\left( x\right) =-\dfrac {6}{\left( 3x+5\right) ^{2}}$
the function is decreasing
So it satisfies all three conditions for integral test
To find whether series is converges or diverges
$$\int ^{\infty }_{1}\dfrac {2}{3n+5}=\int ^{\infty }_{1}\dfrac {2}{3}\dfrac {3}{3n+5}=\dfrac {2}{3}\ln \left( 3n+5\right) ]^{\infty }_{1}=\infty $$
So the series diverges