convergence divergence - How to figure out if the series $sum_{n=1}^infty{ln(frac{n+1}{n+3})}$ is convergent or divergent?

$$\sum_{n=1}^\infty{\ln(\frac{n+1}{n+3})}$$

I have tried every test that we have learned so far in my class.

It's not geometric or telescoping, the Test for Divergence tells me nothing. It has negative terms, so I can't use the Integral Test, the Comparison Test, or the Limit Comparison test. The ratio test doesn't tell me anything.

Those are all the tests we've learned so far in my class. The other thing I've tried is separating it out using properties of logarithms:

$\sum_{n=1}^\infty{\ln(n+1)}-\sum_{n=1}^\infty{\ln(n+3)}$

However, those are two divergent series, and I can't find any theorem in my textbook that says that the sum of two divergent series is divergent.

How can I prove whether this series is convergent or divergent?

Answer

Let $S_k := \sum_{n=1}^{k}\ln\Big(\frac{n+1}{n+3}\Big)$. By induction on $k$, $$\text{if }k\ge 3, \text{then }S_k =\ln\Big(\frac{6}{(k+2)(k+3)}\Big).$$

Hence, $\lim_{k \rightarrow \infty} S_k = -\infty $.