Summation notation from n=1 to n=infinity
sin(n)/n
I believe it converges by the ratio test but my book says it's "inconclusive" because of a comparison to the harmonic series. I don't get it....... any help please?
$\displaystyle
\sum\limits_{n \geqslant 1} {\sin \left( n \right)} < K_{ \in \mathbb{R}} {\text{ }}
$ (you try prove it) and $\displaystyle
\frac{1}
{n}\xrightarrow[{n \to \infty }]{}0
$ for dirichlet, the serie $\displaystyle
\sum\limits_{n \geqslant 1} {\frac{{\sin \left( n \right)}}
{n}}
$ converge
$\displaystyle \sum\limits_{n=1}^{\infty }{a_{n}\sin n}<\infty$ whenever $\displaystyle a_n$ is a decreasing sequence and $\displaystyle \lim_{n\to\infty}a_n=0.$
Here $\displaystyle a_n=\frac1n$ and this fulfills the above conditions, whereat $\displaystyle \sum\limits_{n=1}^{\infty }{\frac{\sin n}{n}}<\infty ,$ and we're done.