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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?

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    Yes, it is. The partial sums diverge to $-\infty$ and hence the series diverges. – 2017-02-24
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    thanks for your help, just wanted to clear that up in my mind – 2017-02-24

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YES.Because the definition of $x=\sum_{j=1}^{\infty}A_j$ is that $x=\lim_{n\to \infty}\sum_{j=1}^nA_j$.