1. Convergence in Series

Hello!

I don't know how to determine whether a series involving a trig function converges or diverges.
As an example, could someone show me how to determine if the following series converges?

$\sum^{\infty}_{n=1} \frac{\left|sin(n)\right|}{n^2}$

I appreciate some guidance. Cheers.

P.S. we don't use the integral test in my course.

2. Use comparison test. Note that: $0 \leq |\sin (n)| \leq 1$

3. Originally Posted by Roam
Hello!

I don't know how to determine whether a series involving a trig function converges or diverges.
As an example, could someone show me how to determine if the following series converges?

$\sum^{\infty}_{n=1} \frac{\left|sin(n)\right|}{n^2}$

I appreciate some guidance. Cheers.

P.S. we don't use the integral test in my course.
Note that |sin(n)| < 1

so,

|sin(n)| / n^2 < 1/n^2

so, does 1/n^2 converge? If so, then your original series converges by the Direct Comparison Test.

4. Originally Posted by coolguy99
Note that |sin(n)| < 1

so,

|sin(n)| / n^2 < 1/n^2

so, does 1/n^2 converge? If so, then your original series converges by the Direct Comparison Test.
Yes 1/n^2 is a convergent p-series with p=2>1, I think I'm starting to get the idea now...

Here's another kind of question that I don't understand:

$\sum^{\infty}_{n=1} sin^2(\frac{1}{n})$

What can we compare this to? We can make use of the fact that sin(n)<1 but I'm not sure what to do next...