Marissa English

2022-01-30

Consider the series

$\sum _{n=1}^{\mathrm{\infty}}\mathrm{log}(1+\frac{1}{\left|\mathrm{sin}\left(n\right)\right|})$

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.

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.

Jaiden Conrad

Beginner2022-01-31Added 14 answers

No, the term $\mathrm{log}(1+\frac{1}{\left|\mathrm{sin}\left(n\right)\right|})$ is not decreasing, but since $\left|\mathrm{sin}\left(x\right)\right|\le 1$ ,it follows that

$\mathrm{log}(1+\frac{1}{\left|\mathrm{sin}\left(n\right)\right|})\ge \mathrm{log}(1+\frac{1}{1})$