consider the series \\( \\sum _ { n = 0 } ^ { \\infty } \\frac { ( x - 8 ) ^ { n } } { 9 ^ { n } } \\)(a)…

consider the series \\( \\sum _ { n = 0 } ^ { \\infty } \\frac { ( x - 8 ) ^ { n } } { 9 ^ { n } } \\)(a) find the series radius and interval of convergence(b) for what values of \\( x \\) does the series converge absolutely?(c) for what values of \\( x \\) does the series converge conditionally?(a) the interval of convergence is \\( ( - 1,17 ) \\) (simplify your answer. type your answer in interval notation)the radius of convergence is 9(b) the series converges absolutely on the interval \\( ( - 1,17 ) \\)(c) for what values of \\( x \\) does the series converge conditionally? select the correct choice below and, if necessary, fill in the answer box to complete your choicea. the series converges conditionally at \\( x = \\)(use a comma to separate answers as needed)b. the series does not converge conditionally

consider the series \\( \\sum _ { n = 0 } ^ { \\infty } \\frac { ( x - 8 ) ^ { n } } { 9 ^ { n } } \\)(a) find the series radius and interval of convergence(b) for what values of \\( x \\) does the series converge absolutely?(c) for what values of \\( x \\) does the series converge conditionally?(a) the interval of convergence is \\( ( - 1,17 ) \\) (simplify your answer. type your answer in interval notation)the radius of convergence is 9(b) the series converges absolutely on the interval \\( ( - 1,17 ) \\)(c) for what values of \\( x \\) does the series converge conditionally? select the correct choice below and, if necessary, fill in the answer box to complete your choicea. the series converges conditionally at \\( x = \\)(use a comma to separate answers as needed)b. the series does not converge conditionally

Answer

Explanation:

Step1: Recall the ratio test formula

For a power series (\sum_{n = 0}^{\infty}a_{n}(x - c)^{n}), we use the ratio test. Here (a_{n}=\frac{1}{9^{n}}), and (a_{n + 1}=\frac{1}{9^{n+1}}). The ratio test formula is (L=\lim_{n\rightarrow\infty}\left|\frac{a_{n + 1}(x - c)^{n+1}}{a_{n}(x - c)^{n}}\right|). Substituting (a_{n}) and (a_{n+1}) and (c = 8) into the formula, we get (L=\lim_{n\rightarrow\infty}\left|\frac{(x - 8)^{n+1}}{9^{n+1}}\cdot\frac{9^{n}}{(x - 8)^{n}}\right|).

Step2: Simplify the limit

Simplify the expression inside the limit: (L=\lim_{n\rightarrow\infty}\left|\frac{x - 8}{9}\right|=\left|\frac{x - 8}{9}\right|). For convergence, (L<1). Solving (\left|\frac{x - 8}{9}\right|<1) gives (- 1<\frac{x - 8}{9}<1). Multiply each part of the inequality by (9): (-9<x - 8<9). Then add (8) to each part: (-1<x<17). The radius of convergence (R) is found from (\left|\frac{x - 8}{9}\right|<1), and (R = 9).

Step3: Check endpoints for absolute and conditional convergence

  • When (x=-1): The series becomes (\sum_{n=0}^{\infty}\frac{(-1 - 8)^{n}}{9^{n}}=\sum_{n = 0}^{\infty}\frac{(-9)^{n}}{9^{n}}=\sum_{n=0}^{\infty}(- 1)^{n}). This series diverges (by the divergence test, (\lim_{n\rightarrow\infty}(-1)^{n}) does not exist).
  • When (x = 17): The series becomes (\sum_{n=0}^{\infty}\frac{(17 - 8)^{n}}{9^{n}}=\sum_{n=0}^{\infty}\frac{9^{n}}{9^{n}}=\sum_{n=0}^{\infty}1). This series diverges (by the divergence test, (\lim_{n\rightarrow\infty}1\neq0)).

Since the series does not converge at the endpoints (x=-1) and (x = 17), and for absolute convergence we have the same interval as the interval of convergence (because the ratio - test gives absolute convergence for (L<1)), and there are no points where the series converges conditionally (because conditional convergence can only occur at the endpoints of the interval of convergence for a power series, and our endpoints do not give convergence).

Answer:

(b) The series converges absolutely on the interval ((-1,17)) (c) B. The series does not converge conditionally.