Convergent or Divergent

Question

Determine if the series converges or diverges: $$\sum_{n=1}^\infty \frac{\ln(n)}{n \left|\sin(n)\right|} = $$

We know that

$$ \sum_{n=1}^\infty \frac{1}i = \text{Divergent} $$

$$ \sum_{n=1}^1 \ln(i )= 0 $$

How to solve this?

Answer 1

Another way: also note $\frac{\log n }{n|\sin n|} \geq \frac{\log n}{n}$. Now look at $\sum_{n=1}^{\infty}\frac{\log n }{n}$. By intergral test if the integral diverges, then the sum diverges too. $\int_{1}^{\infty} \frac{\log x dx}{x} = \left. \frac{ \log^{2} x}{2} |^{\infty}_{1}\right.=\infty$. The integral diverges, hence the sum diverges too. Since the original sum is lower-bounded by this one, by comparison test it diverges too.

Answer 2

Hint: Since $|\sin n|\le 1$, and $\ln n \gt 1$ for $n\ge 3$, the $n$-th term of our sequence is greater than $\frac{1}{n}$ if $n\ge 3$. Now use the Comparison Test.