a convergence/divergence question (i've been going at it for two days now . . . .)

enormousface · #1 $Old$ May 15th, 2009, 09:01 AM

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

chisigma · #2 $Old$ May 15th, 2009, 09:58 AM

From geometry it is well known that for $0<x<\frac{\pi}{2}$ ...

$\sin x < x < \tan x$ (1)

Now from (1) we derive that...

$\frac{\sin \frac{1}{n}}{n}<\frac {1}{n^{2}}$

The series...

$\sum_{n=1}^{\infty} \frac{1}{n^{2}}$

... converges so that also converges the series...

$\sum_{n=1}^{\infty} \frac{\sin \frac{1}{n}}{n}$

Kind regards

$\chi$ $\sigma$