I used a limit comparison test, what do I do next

Test for convergence/divergence?
This is from the 6th edition of Calculus by James Stewart, page 745, #9.

Determine whether the series converges of diverges. I will use S to denote the sum from 1 to infinity.

S(cos^2(n))/(n^2+1).

I decided to use a limit comparison test.

a_n = S(cos^2(n))/(n^2+1)
b_n = 1/(n^2 + 1)

First, I used the integral test to determine whether b_n was convergent or not, and it is convergent (specifically to pi/2).

Then I used the limit comparison test on the two (all limits I write that tend to infinity, I'll use write with "lim")

lim a_n / b_n = lim (cos^2(n)/(n^2+1)) / (1/(n^2 + 1)) = lim cos^2(n).

Then I don't know what to do. I know that the cos function is oscillating, so wouldn't cos^2(n) also be oscillating, meaning that the series doesn't converge to anything? Can somebody please help me and explain how to get to the solution? The answer is convergent.

Thank you.

The Limit Comparison Test not apply with the function you chose.

However, we can use the Direct Comparison Test.
Since cos^2(n)/(n^2 + 1) ≤ 1/(n^2 + 1) for all n > 0.

Since Σ(n = 1 to ∞) 1/(n^2 + 1) is convergent (from the integral test, as you said),
the series in question must also converge.

I hope this helps!

Since -1 ≤ cos n ≤ 1 for all n, we have 0 ≤ cos^2(n) ≤ 1 for all n.

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