Does the Series ∑ n*sin(1/n) from n=1 to Infinity Converge?

[rcmango]

Homework Statement

Does the series infinity n(sin(1/n)) converge?
E
n=1

Homework Equations

n

and

sin(1/n)

The Attempt at a Solution

The equation sin(1/n) looks familiar, maybe i could use the squeeze theorem?

something like -1 <= sin(1/n) <= 1

i'm not sure where to go after this though, or if I'm even on the right track?
please help.
thankyou.

 

[arildno]

First check:
Does the general term go to zero as n goes to infinity?

 

[Gib Z]

Yes You could use the squeeze theorem and get -|x|< sin(1/x) < |x|, but that doesn't seem the help you much. Do what arildno said, maybe a few other convergence tests, try using Taylor Series? I'd say it does, but can't show you right now.

 

[dextercioby]

Since this is not an alternating series, then only answering the question posed by Arildno is enough to solve the problem.

Daniel.

 

[lkh1986]

There is a general formula: as x approaches 0, (sin x)/x approaches 1. So you can apply this in your question. Let n=1/x, then you modify the equation, then you will get the answer that the series diverge, since the series does not have a sum.

 

[rcmango]

read next post.

 

[rcmango]

First check:
Does the general term go to zero as n goes to infinity?

yes. i would say as n goes to infinity, it approaches 0.
the reasoning is because starting with small numbers it gets larger around .01745...

don't know where to go from here.

thanks for all the feedback, i found it all useful.

 

[d_leet]

yes. i would say as n goes to infinity, it approaches 0.
the reasoning is because starting with small numbers it gets larger around .01745...

don't know where to go from here.

thanks for all the feedback, i found it all useful.

Numbers at a certain point don't really tell you much about the behavior at infinity. Take the limit of the terms as n goes to infinity, if it isn't 0 what can you say about the series?

 

[rcmango]

Numbers at a certain point don't really tell you much about the behavior at infinity. Take the limit of the terms as n goes to infinity, if it isn't 0 what can you say about the series?

that it diverges.

 

[theperthvan]

that it diverges.

yes it does