Determine if series converges or diverges

[turbokaz]

Homework Statement

Ʃ cos^2(n)/(n^2+8)

Ʃ 5n/(n^2+1) * cos(2πn)

Homework Equations

The Attempt at a Solution

I think that both series diverge. Can anyone validate this or tell me if I'm wrong?

 

[kru_]

Why do you think they diverge?

 

[turbokaz]

I'm changing my mind. First one diverges because the cosine will oscillate between -1 and 1. The second one converges because it will always go to 0?

 

[Dansuer]

Convergence and divergence of infinite series is a counterintuitive and complicated matter. Don't base you're conclusion of guesses and intuition. Use theorems instead. Do you know any? Which ones may be useful for those series?

 

[Bohrok]

I'm changing my mind. First one diverges because the cosine will oscillate between -1 and 1. The second one converges because it will always go to 0?

You have cos²n which will always be positive since it's squared. As a result, cos²n ≤ 1 and so cos²n/(n² + 8) ≤ 1/(n² + 8). From there it's not too difficult to show whether it converges or diverges using one of the series tests.

Before looking at the second one, which series tests are you familiar with?