Answer
The series is divergent
Work Step by Step
$\Sigma^{\infty}_{n=1} \frac{n}{1+\sqrt n}$
$a_{n}= \frac{n}{1+\sqrt n}$
partial sum $s_{n} = a_{1} + a_{2} +... a_{n}$
$n=1$ $a_{1}=0.500$ $s_{1}=0.500$
$n=2$ $a_{2}=0.82843$ $s_{2}=1.3284$
$n=3$ $a_{3}=1.09808$ $s_{3}=2.4265$
$n=4$ $a_{4}=1.3333$ $s_{4}=3.7598$
$n=5$ $a_{5}=1.54508$ $s_{5}=5.3049$
$n=6$ $a_{6}=1.73939$ $s_{6}=7.0443$
$n=7$ $a_{7}=1.92004$ $s_{7}=8.9644$
$n=8$ $a_{8}=2.08963$ $s_{8}=11.0540$
The partial sum appears to be increasing without bound so the series is divergent.