Does the series \sum_{n=1}^\infty sin(\frac{1}{n^2}) converge?

[NDakota]

Homework Statement

Does the follow serie converge:
$$\sum_{n=1}^\infty sin(\frac{1}{n^2})$$

Homework Equations

For serie $$a_n$$ and $$b_n$$ if:

A = $$0 \leq a_n \leq b_n$$

if $$b_n$$ converges then $$a_n$$ converges

The Attempt at a Solution

I think that I have to use the equation (see 2) and then with

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

I think that it is larger than A. However I need proof... Any suggestions.

Thanks in advance.

 

[morphism]

For x>=0, |sin(x)| <= x. (Better yet, on [0,1], 0 <= sin(x) <= x.) Or you can just use the limit comparison test.

Note that both convergence tests require your series to have nonnegative terms.