Final answer:
The series ∑¹∞ (4/3)ⁿ is determined to be divergent using the ratio test, as the constant ratio of consecutive terms is 4/3, which is greater than 1.
Step-by-step explanation:
To determine if the series ∑¹∞ (4/3)ⁿ is convergent or divergent, we can use the ratio test. The ratio test states that for a series ∑ aᵢ, if the limit L = lim |aᵢ₊₁/aᵢ| as i approaches infinity exists and is less than 1, the series is convergent. If L is greater than 1, or if the limit does not exist, the series is divergent. In the case where L equals 1, the test is inconclusive.
To use the ratio test on our series, we calculate the limit of the absolute value of the ratio of consecutive terms.
Let aᵢ = (4/3)ⁿ. Then, the ratio of aᵢ₊₁/aᵢ is: |aᵢ₊₁/aᵢ| = |(4/3)ᵢ⁺¹ / (4/3)ᵢ| = 4/3. Since the ratio is a constant and does not depend on i, we don't need to find the limit; it is already the constant value of 4/3, which is greater than 1. Therefore, by the ratio test, the series is divergent.
Since the magnitude of the ratio is greater than 1, the terms of the series get larger as n increases, which implies that the sum of the terms will also grow infinitely large, leading to divergence.