Does the series \sum^{\infty}_{n=1}sin(\frac{1}{n^{4}}) converge?

[Bipolarity]

Homework Statement

Determine whether the following series diverges, converges conditionally, or converges absolutely.

$$\sum^{\infty}_{n=1}sin(\frac{1}{n^{4}})$$

Homework Equations

The Attempt at a Solution

This was on today's test, and was the only problem I wasn't able to solve. I doubt my teacher will be going over these, and in any case his explanations never satisfy me, so could someone help me with this?

According to the nth term test, the limit of the sequence is 0, since sin(x) is continuous, so the function doesn't necessarily diverge.

Integral test can't be applied because the sequence is not monotonic. Root test serves no purpose. Limit comparison might work, but with what? Ratio does not work (I think?).

How might I approach this?

BiP

 

[STEMucator]

Hedi beat me to it down below VVV haha. I forgot that something bigger that diverges tells you nothing.

 

[hedipaldi]

use limit comparison with 1/n^4.for large n this is a positive series.

 

[Bipolarity]

use limit comparison with 1/n^4.for large n this is a positive series.

Thanks!

But damn! I wish I thought of that in the test! Oh well.

BiP