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.
Best Answer
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?