Determine whether the series [infinity] cos(n) n2 n = 1 = cos(1) 12 + cos(2) 22 + cos(3) 32 + ⋯ is convergent or divergent. SOLUTION The series has both positive and negative terms, but it is not alternating. (The first term is positive, the next three are negative, and the following three are positive: The signs change irregularly.) We can apply the Comparison Test to the series of absolute values [infinity] cos(n) n2 n = 1 = [infinity] |cos(n)| n2 n = 1 . Since |cos(n)| ≤ 1 Correct: Your answer is correct. for all n, we have |cos(n)| n2 ≤ 1 n2 . We know that Σ 1/n2 is convergent p−series with p = and therefore Σ |cos(n)|/n2 is convergent by the Comparison Test. Thus the given series Σ (cos(n))/n2 is absolutely convergent and therefore convergent by this theorem.