i've been thinking about this problem for almost two days, and i havent made much headway
1.) does the following series converge?
[epsilon]infiniti,n=1 (sin(1/n))/n
From geometry it is well known that for $\displaystyle 0<x<\frac{\pi}{2}$...
$\displaystyle \sin x < x < \tan x$ (1)
Now from (1) we derive that...
$\displaystyle \frac{\sin \frac{1}{n}}{n}<\frac {1}{n^{2}} $
The series...
$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{2}}$
... converges so that also converges the series...
$\displaystyle \sum_{n=1}^{\infty} \frac{\sin \frac{1}{n}}{n}$
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$