Answer
Function satisfies conditions for integral test
Series diverges
Work Step by Step
$\dfrac {\ln 2}{2}+\dfrac {\ln 3}{3}+\dfrac {\ln 4}{4}+\dfrac {\ln 5}{5}+\dfrac {\ln 6}{6}\ldots =\sum ^{\infty }_{n=2}\dfrac {\ln n}{n}$
In order for integral test to be applied $f\left( x\right) =\dfrac {\ln x}{x}$ must be positive continuous and decreasing for $x>1$
We see that function is positive and continuous
$f'\left( x\right) =\dfrac {1-\ln x}{x^{2}} $
the function is decreasing
So integral test can be applied
$$ \int ^{\infty }_{1}\dfrac {\ln x}{x}=\dfrac {1}{2}\left( \ln x\right) ^{2}]^{\infty }_{1}=\infty $$
The integral diverges, so the series diverges