Test for Convergence ?

Question

The sum (from n=1 to infinity) of 1/(n+n(cos(n))^2)

I tried using the Limit Comparison Theorem, but I get 1 and 1/2 since cosine squared varies between 0 and 1 indefinitely.

I'm not sure if that's enough to show that the summation is convergent.

Thanks in advance.

Answer 1

As the harmonic series Σ [i=1 to ∞] 1/n diverges, it is elementary to prove that Σ [i=1 to ∞] 1/(2n) diverges, and as 1/(n+n(cos(n))^2) ≥ 1/(2n), it follows by the comparison test that Σ [i=1 to ∞] 1/(n+n(cos(n))^2) diverges as well.