query: Prove that $\sum_{n=1}^{\infty}\frac{|\cos n|}{n}$ diverges
positive_passages.docid: 149850
positive_passages.text: Methods for determining the convergence of $\sum\frac{\cos n}{n}$ or $\sum\frac{\sin n}{n}$
negative_passages.docid: 145019
negative_passages.text: Prove that $\sum_{n=0}^\infty \frac{(-1)^n}{3n+1} = \frac{\pi}{3\sqrt{3}}+\frac{\log 2}{3}$
negative_passages.docid: 162
negative_passages.text: Determine the convergence of $ \sum_{n=1}^{\infty}\left[1-\cos\left(1 \over n\right)\right] $
negative_passages.docid: 75459
negative_passages.text: study the convergence of $\sum_{n=1}^\infty \log \frac{n+1}{n}$
negative_passages.docid: 7198
negative_passages.text: How can I show whether the series $\sum\limits_{n=1}^\infty \frac{(-1)^n}{n(2+(-1)^n)} $ converges or diverges?
negative_passages.docid: 101138
negative_passages.text: Convergent or divergent: $\sum_{k=1}^{\infty}\frac{2^{k}\cdot k!}{k^{k}}$
negative_passages.docid: 72377
negative_passages.text: Show that $\sum_{n=1}^\infty r^n\cos(n\theta)=\dfrac{r\cos\theta -r^2}{1-2r\cos\theta+r^2}$ whenever $0<r<1$.
negative_passages.docid: 149240
negative_passages.text: Divergence of a series
negative_passages.docid: 9263
negative_passages.text: Convergence of $\sum_{n=1}^\infty\frac{n!\, i^n}{n^n}$
negative_passages.docid: 77183
negative_passages.text: Prove that $ \sum_{n=1}^\infty \ln\big(n\sin \frac{1}{n}\big)$ converges.
negative_passages.docid: 74286
negative_passages.text: Prove: $\lim_{n\to\infty} \frac{\sum^n_{k=1} \frac{1}{k}}{ \ln(n) } = 1$
negative_passages.docid: 132613
negative_passages.text: Testing the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \cos{n}}}$
negative_passages.docid: 725
negative_passages.text: proof of $\sum_{n=1}^\infty n \cdot x^n= \frac{x}{(x-1)^2}$
negative_passages.docid: 63380
negative_passages.text: Show that the series $\sum_{n=2}^{\infty} \frac{1}{n\, (\log n)^s}$ converges for $s >1$ and diverges for $s=1$
negative_passages.docid: 25503
negative_passages.text: Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$
negative_passages.docid: 119147
negative_passages.text: How to prove that $\sum_{n=1}^{\infty}\frac{|\sin{(n\theta_0)}|}{n}$ diverges for given $\theta_0\in (0,\pi)$?
negative_passages.docid: 12480
negative_passages.text: Convergence of $\sum_{n=1}^{\infty} \frac{\frac{1}{2}+(-1)^n}{n}$