I am somewhat a noob, and I don’t recall my math preparation from college. I know that the sum $\sum_{n \geq 1} \frac{1}{n}$ is divergent and my question is if the sum $$\sum_{n \geq 1} \frac{2^n\operatorname{mod} n}{n^2}$$ converges. I think is not but I do not know how to prove that! Thanks!

If I have $\sum_{n=2}^{\infty} \frac {1}{n\log n}$ and want to prove that it diverges, can I use following? $$\frac {1}{n\log n} \lt \frac {1}{n}$$ $\sum_{n=1}^\infty \frac 1 n$ diverges, but the limit of $\frac 1 n$ equals to zero so the comparasion I think isn’t useful. Or can I say it diverges because $\sum_{n=1}^\infty \frac […]

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