Question

find the radius of convergence, r, of the series. [infinity] n 7n (x 4)n n = 1 r = find the interval, i, of convergence of the series. (enter your answer using interval notation.) i =

Answer 1

To find the radius of convergence of the series [infinity] n 7n (x 4)n n = 1, we can use the ratio test. The limit of the ratio of consecutive terms simplifies to (x+4)/(x+4), which is equal to 1 for any value of x. Therefore, the radius of convergence is ∞ and the series converges for all x.

Step-by-step explanation:

To find the radius of convergence of a power series, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms of the series is less than 1, then the series converges. In this case, we have the series [∞] n 7n (x 4)n n = 1. To apply the ratio test, we need to find the limit of the absolute value of the ratio of consecutive terms:

lim |((n+1) 7(n+1) ((x+4)^(n+1))) / (n 7n (x+4)^n)| as n approaches infinity.

Simplifying this expression, we have:

lim |(n+1)/(n) * (7/(7)) * ((x+4)/(x+4)) * ((x+4)^n/(x+4)^(n+1))| as n approaches infinity.

→ lim |(x+4)/(x+4)| as n approaches infinity.

= |(x+4)/(x+4)|.

Since this limit is equal to 1 for any value of x, the radius of convergence is ∞, which means the series converges for all values of x.

Answer 2

To determine the radius of convergence, r, and the interval of convergence, i, for the series, we apply the Ratio Test. The radius of convergence is found to be r = 1/7, and the interval of convergence is i = (4 - 1/7, 4 + 1/7).

Step-by-step explanation:

To find the radius of convergence, r, for the given power series ∑ n 7n (x - 4)n from n = 1 to infinity, we can use the Ratio Test. Applying the Ratio Test involves taking the limit of the absolute value of the ratio of successive terms as n approaches infinity:

L = lim (n -> ∞) |an+1 / an|

If L < 1, the series converges absolutely. In this case, we have:

L = lim (n -> ∞) |(n+1) * 7n+1 * (x - 4)n+1 / (n * 7n * (x - 4)n)|

After simplifying, we get:

L = lim (n -> ∞) |7(x - 4)|

Since L must be less than 1 for convergence, we have |7(x - 4)| < 1, which yields |x - 4| < 1/7.

The radius of convergence is r = 1/7. The interval of convergence, i, will be within the range where |x - 4| < 1/7, thus i = (4 - 1/7, 4 + 1/7) in interval notation.