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

- May 15th 2009, 08:01 AMenormousfacea convergence/divergence question (i've been going at it for two days now . . . .)
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 - May 15th 2009, 08:58 AMchisigma
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

$\displaystyle \chi$ $\displaystyle \sigma$