1. ## Series/convergence problem

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?

2. $
\sum\limits_{n \geqslant 1} {\sin \left( n \right)} < K_{ \in \mathbb{R}} {\text{ }}
$
(you try prove it) and $
\frac{1}
{n}\xrightarrow[{n \to \infty }]{}0
$
for dirichlet, the serie $
\sum\limits_{n \geqslant 1} {\frac{{\sin \left( n \right)}}
{n}}
$
converge

3. $\sum\limits_{n=1}^{\infty }{a_{n}\sin n}<\infty$ whenever $a_n$ is a decreasing sequence and $\lim_{n\to\infty}a_n=0.$

Here $a_n=\frac1n$ and this fulfills the above conditions, whereat $\sum\limits_{n=1}^{\infty }{\frac{\sin n}{n}}<\infty ,$ and we're done.

4. Originally Posted by Kaitosan
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?
That's very strange. You can say that a particular test for convergence is "inconclusive" but that doesn't apply to a series itself! Obviously, any given series either converges or it doesn't. That's true whether we know which is true or not!