Alternating Series Convergence: Is the Series Sum of sin(1/n^2) Convergent?

[CalculusHelp1]

Homework Statement

Sorry I don't know how to use symbols on this site so bear with me:

the question is does the following series converge: sum of sin(1/n^2) where n goes from 1 to infinity

Homework Equations

Limit comparison test, maybe others

The Attempt at a Solution

Okay I think I got the solution but I'm not sure if the logic is correct. I divided sin(1/n^2) by the p-series (1/n).

This comes out to n/sin(1/n^2)...the top will go to infinity and the bottom will go to 0 as n goes to infinity, so the series divergies. Because 1/n also diverges (harmonic series) sin(1/n^2) also diverges. Is this right?

 

[CalculusHelp1]

Don't know why I said 'alternating' series in the thread title...my brain is becoming mush!

 

[thegreenlaser]

I think limit test is probably the way to go, however I think you're doing a few things wrong with it.

With $$a_n$$ being the given function, consider $$b_n = \frac{1}{n^2}$$

Then,
$$\lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} n^2\sin{\frac{1}{n^2}}$$

Now, you should be able to show that that limit is finite, so the convergence of the sum of $$b_n$$ would imply the convergence of the sum of $$a_n$$. Does that help?

 

[CalculusHelp1]

Yep that fixes it, I also made an algebraic error thinking sin was on the denominator. Brain is going to mush. Thanks!