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 $f\left( x\right) =\dfrac {1}{x+3}$ is positive and continuous for $x \geq1$ and in order to find if the function is decreasing we need to find derivative of function
$f'\left( x\right) =-\dfrac {1}{\left( x+1\right) ^{2}}$
the function is decreasing
So it satisfies all three conditions for integral test
To find whether series is converges or diverges
$$\int ^{\infty }_{1}\dfrac {1}{x+3}=\ln \left( x+3\right) ] ^{\infty }_{1}=\infty $$
So the series diverges