Question

suppose that [infinity] cn x n n = 0 converges when x = −4 and diverges when x = 6. what can be said about the convergence or divergence of the following series?

Answer 1

The power series converges at x = -4 and diverges at x = 6, indicating a radius of convergence somewhere between 4 and 6. The series must converge for all x within (-4, 4) by the Comparison Test, but additional information is needed to determine convergence outside this interval.

Step-by-step explanation:

The question seems to involve a power series of the form ∑ cn x^n from n = 0 to infinity, with convergence provided for x = -4 and divergence at x = 6. To discuss the convergence or divergence of a new series, we must consider the interval of convergence for the original series. If a series converges at x = -4 and diverges at x = 6, this suggests that the radius of convergence, R, is less than 6 but greater than or equal to 4. Hence, the interval of convergence lies somewhere within the interval (-R, R). Without additional information about the specific behavior of the series at the endpoints, we cannot give a definitive conclusion on the convergence of the series at other values.

Furthermore, as the series converges when x = -4, by the Comparison Test, the series must also converge for all x within the interval (-4, 4), since those values are closer to 0 and should therefore produce smaller terms in the series assuming the coefficients cn do not grow too quickly in magnitude.

Answer 2

The series converges when -4 < x < 6 and diverges outside this interval.

Step-by-step explanation:

The question states that the series [infinity] cn x n n = 0 converges when x = -4 and diverges when x = 6. We need to determine the convergence or divergence of the series. For the series to converge, the sum of the terms must approach a finite value as n goes to infinity. Since the series converges at x = -4, we can conclude that the series converges when -4 < x < 6. Outside this interval, the series diverges.