Infinite Series problem with cos involved.

[salazar888]

Homework Statement

I have to determine whether the series converges or diverges.

\sum (cos^2 (n)) / n^2 +1

Homework Equations

Suppose An and Bn are series with positive terms. If the limit of An over Bn as n approaches infinity equals C, and C is a finite number greater than 0, then both series are coupled and they both either converge or diverge.

The Attempt at a Solution

First I let An to be cos^2 n / n^2 + 1 and Bn = cos^2 n / n^2. Therefore, Bn is greater than An for all n. I first used the sequence divergence test obtaining zero, meaningless. Then I used the Limit Comparison Test and the result was 1, which is greater than 0, meaning that they are coupled. Then I was trying to find the improper integral of cos^2 x / x^2, but I'm stuck because I don't really have anything to substitute there. I can't remember solving any integral like this one. I was wondering what other method I could use at this point. I know the series converges but I need to prove that. The only information I have from the limit comparison test is that my An and Bn are coupled. Should I find another Bn? Thank you.

 

[salazar888]

I just realized I don't even meet the requirements to do integral test since cosine is not a continuous function. I apologize. I figure I have to change the Bn then. I would have to find a larger function than An that converges.

 

[icystrike]

Although i don't get what ur talking about, may i ask u what is the upperbound of cos^{2}(n)?

 

[salazar888]

1?

 

[salazar888]

ok Thanks I get it now. I didn't think I could use 1 / n^2 as my greater function. Then I can just use p-series at this point. I was trying to be as clear as possible, I've only been on the website for a couple of days.