Question

Find all the values of x such that the given series would converge. (8x - 6)" n?

Find all the values of x such that the given series would converge. Σ N=1

Answer 1

The values of x for which the series converges are those in the interval (5/8, 7/8), determined by applying the ratio test for convergence of power series.

Step-by-step explanation:

The student is asking for the range of values for x that will make the series converge. This series is in the form of a power series, where the terms are (8x - 6)n to the power of n. To determine the convergence of this series, we utilize the ratio test for convergence of power series. The ratio test states that if the absolute value of the ratio between the n+1 term and the n term is less than 1 as n approaches infinity, the series converges. Applying this to our series:

1. Find the ratio |(8x - 6)n+1/(8x - 6)n|.
2. Simplify the ratio to |8x - 6|.
3. For convergence, set the inequality |8x - 6| < 1.
4. Solve for the values of x: 5/8 < x < 7/8.

The series converges for x in the interval (5/8, 7/8).