I have to determine whether the series converges or diverges.
[itex]\sum (cos^2 (n)) / n^2 +1 [/itex]
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.