Converge? $\sum_{k=1}^{\infty}\frac{ \sin \left(\frac{1}{k}\right) }{k} $

Question

Determine whether the series converges: $$\sum_{k=1}^{\infty}\frac{ \sin \left(\frac{1}{k}\right) }{k} $$

I tired to use direct comparison test but since 1/k is not convergent, so I am not sure about in this case.

score 8 · Accepted Answer · answered Mar 15 '16 at 18:24

8

Hint

Since you want a direct comparison use this inequality

$$\sin(x)\le x,\; \forall x\ge0$$

answered Mar 15 '16 at 18:24

user296113

7,570

score 3 · Answer 2 · answered Mar 15 '16 at 18:24

3

It converges absolutely: $\left|\sin\left(\dfrac{1}{k}\right)\right| < \dfrac{1}{k}$

answered Mar 15 '16 at 18:24

DeepSea

77,651

Bernard · Answer 3 · 2016-03-15T18:37:52.160

2

It converges since it has positive terms and: $$ \sin\dfrac1k \sim_{\infty}\frac1k,\enspace\text{hence}\enspace\frac{ \sin\frac1k }{k}\sim_{\infty}\frac1{k^2}.$$

edited Mar 15 '16 at 18:37

answered Mar 15 '16 at 18:30

Bernard

175,478

It's useless to say that it converges absolutely since it's a positive series. – user296113 Mar 15 '16 at 18:33
An old habit. It may happen the first terms of a series are not positive., even if they're eventually positive. – Bernard Mar 15 '16 at 18:36

Converge? $\sum_{k=1}^{\infty}\frac{ \sin \left(\frac{1}{k}\right) }{k} $

3 Answers3