Final answer:
To determine if the series converges or diverges, apply the ratio test, which states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, the series converges. In this case, the series converges with a sum of 40/9.
Step-by-step explanation:
To determine whether the series ∑(n=2 to [infinity]) (5 * 2^(n-1)) / (3^(n+1)) converges or diverges, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms of a series is less than 1, then the series converges. Let's apply the ratio test to this series:
First, let's find the ratio of consecutive terms:
|(5 * 2^n) / (3^(n+1)) / (5 * 2^(n-1)) / (3^n)| = |2/3| = 2/3
Since the ratio is less than 1, the series converges. To find its sum, let's use the formula for a geometric series:
S = a / (1 - r)
where 'a' is the first term and 'r' is the common ratio. In this case, the first term is (5 * 2) / (3^3) = 40 / 27 and the common ratio is 2/3. Plugging these values into the formula, we get:
S = (40 / 27) / (1 - 2/3) = (40 / 27) / (1/3) = (40 / 27) * (3/1) = 40/9
Therefore, the sum of the series ∑(n=2 to [infinity]) (5 * 2^(n-1)) / (3^(n+1)) is 40/9.