Answer:
Absolutely convergent
Divergent
Absolutely convergent
Step-by-step explanation:
│sin(4n) / 4ⁿ│is less than or equal to 1 / 4ⁿ for all n ≥ 1. 1 / 4ⁿ is a geometric series with r = 1/4. Therefore, 1 / 4ⁿ converges, so aₙ is absolutely convergent.
(3 − cos(6n)) / (n^⅔ − 2) is greater than 1/n for all n ≥ 3. 1/n diverges, so the series also diverges.
2 + cos(n) is at most 3, so the ratio (2 + cos(n)) / √n is between 0 and 1 for all n greater than 9. So there exists a number r between 0 and 1 such that (2 + cos(n)) / √n ≤ r for n ≥ 9. Since this is a geometric series that converges, the series also converges.
