Determine whether the given series is absolutely convergent or conditionally convergent

Question

Consider the series $$\sum_{n=1}^\infty \log\left(1+\frac{1}{|\sin(n)|}\right).$$ Determine whether it converges absolutely or conditionally.

I am trying to apply Cauchy condensation test, but I am not sure whether the given series is non-increasing or not.

Answer 1

No, the term $\log\left(1+\frac{1}{|\sin(n)|}\right)$ is not decreasing, but since $|\sin(x)|\leq 1$, it follows that $$ \log\left(1+\frac{1}{|\sin(n)|}\right)\geq \log\left(1+\frac{1}{1}\right).$$ Can you take it from here?