Convergence of sin(1/n) Summation from n=1 to Infinity

[Frillth]

Okay, this is my last problem today. I swear.

Homework Statement

For this problem, I need to find the convergence of the sum of sin(1/n) from n=1 to infinity.

Homework Equations

None

The Attempt at a Solution

I know that this has to converge, but I'm having a hard time proving it. It seems like I should either be using the limit comparison test or just the comparison test, but I can't think of whta I can compare it to.

 

[Frillth]

This is very counter-intuitive. I just tried the limit comparison test with 1/n as follows:

lim sin(1/n) = cos(1/n) * -1/n^2 = cos(1/n) = 1
n->inf 1/n = -1/n^2

That means that sin(1/n) must diverge. Is this right?

 

[StatusX]

I don't follow anything from your second post, except the conclusion, which is correct. But please describe more clearly how you arrived at it.