Final answer:
To find the values of x for which the series converges, we apply the convergence criterion for geometric series to the given expression (8x - 6)^n, resulting in an interval (5/8, 7/8) for x.
Step-by-step explanation:
The question is asking to determine the range of values for x that would make the series converge. The series provided, (8x - 6)^n, is a geometric series where the common ratio is (8x - 6). For a geometric series to converge, the common ratio must have an absolute value less than 1. The condition for convergence can be written as:
|8x - 6| < 1
Solving for x gives us two inequalities:
- 8x - 6 < 1
- -(8x - 6) < 1
From the first inequality, solving for x yields:
8x < 7
x < 7/8
From the second inequality, we get:
8x - 6 > -1
8x > 5
x > 5/8
Therefore, for the series to converge, x must be in the interval (5/8, 7/8).