Given the following series,
$$\sum_{n=1}^{\infty} \ln\left(\frac{n}{n+2}\right)$$
I am trying to prove if it diverges or converges.
I used the partial sum test and found that the nth partial sum of this series is,
$$\ln(2)-\ln(n+1)-\ln(n+2)$$
Therefore taking the limit of this partial sum to see if it converges to a number we see that it converges to infinity.
So is this enough to prove that the above series diverges by the nth parial sum test?