Answer: the series converges for x in the interval (7/3, 3).
Explanation:
The given series is a summation of the form: ∑(n = 1 to ∞) ((3x - 8)^n / n^2)
This series converges when the absolute value of the ratio, |(3x - 8)|, is less than 1, as it resembles a power series with a common ratio (3x - 8).
To determine the values of x for which the series converges, we need to solve the inequality:
|3x - 8| < 1
This inequality can be broken down into two separate inequalities:
-1 < 3x - 8 < 1
Adding 8 to all parts of the inequality, we get:
7 < 3x < 9
Dividing all parts by 3:
7/3 < x < 3
So, the series converges for x in the interval (7/3, 3).