#### Answer

The series is divergent.

#### Work Step by Step

$\Sigma^{\infty}_{n=1} \frac{1}{\ln (n+1)}$
$a_{n}=\frac{1}{\ln (n+1)}$
partial sum $s_{n} = a_{1} + a_{2} + ... +a_{n}$
$n=1$ $a_{1}=1.44270$ $s_{1}=1.4427$
$n=2$ $a_{2}=0.91024$ $s_{2}=2.3529$
$n=3$ $a_{3}=0.72135$ $s_{3}=3.0743$
$n=4$ $a_{4}=0.62133$ $s_{4}=3.6956$
$n=5$ $a_{5}=0.55811$ $s_{5}=4.2537$
$n=6$ $a_{6}=0.51390$ $s_{6}=4.7676$
$n=7$ $a_{7}=0.48090$ $s_{7}=5.2485$
$n=8$ $a_{8}=0.45512$ $s_{8}=5.7036$
The series appears to be divergent because it is not increasing to any particular number.