Marissa English

2022-01-30

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

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

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