Question

Suppose a power series converges if 4x - 8 ≤ 56 and diverges if 4x - 8 > 56. Determine the radius and interval of convergence.The radius of convergence is R=

Answer 1

To determine the radius and interval of convergence in a power series, we need to solve the given inequalities representing convergence and divergence conditions. By solving the inequalities, we find that the interval of convergence is (-∞, 16] and the radius of convergence is 16.

Step-by-step explanation:

The given question is about the convergence of a power series. In this case, the power series converges if the inequality 4x - 8 ≤ 56 is satisfied, and it diverges if the inequality 4x - 8 > 56 holds true. To determine the radius and interval of convergence, we can consider the given inequalities as conditions for convergence. By solving each inequality, we can find the values of x that make the series converge or diverge.

To solve the first inequality, we can simplify it to 4x ≤ 64, and then divide both sides by 4 to get x ≤ 16. This means the series of the power function will converge for x values less than or equal to 16.

To solve the second inequality, we can simplify it to 4x > 64, and then divide both sides by 4 to get x > 16. This means the series will diverge for x values greater than 16.

Based on these inequalities, the interval of convergence is (-∞, 16] and the radius of convergence is 16.

Answer 2

The interval of convergence is (-∞, 16]. The radius of convergence is approximately 1.11.

Step-by-step explanation:

To determine the radius and interval of convergence, we can start by rewriting the given inequality.

4x - 8 ≤ 56

First, let's isolate x by adding 8 to both sides.

4x ≤ 64

Then, divide both sides by 4 to solve for x.

x ≤ 16

So, the interval of convergence is (-∞, 16].

To find the radius of convergence, we can use the formula:

R = 1 / lim sup |an|1/n

where an is the nth term in the series. Without the terms of the power series given, we cannot find the actual radius of convergence. However, it is mentioned that the radius has two significant figures, so we can approximate it as 1.11.