Determine the convergence of the series log (1 + 1/|sin n|)

by RoRi

March 11, 2016

Consider the series

$\sum_{n=1}^{\infty} \log \left( 1 + \frac{1}{|\sin n|} \right).$

Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.

The series diverges.

Proof. Since $|\sin n| \leq 1$ for all $n$ , and the logarithm is an increasing function, we have

$\log \left( 1 + \frac{1}{|\sin n|} \right) \geq \log 2$

for all $n$ . Hence, the terms are not going to 0, so the series diverges $. \qquad \blacksquare$

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