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

consider the series \\( \\sum _ { n = 0 } ^ { \\infty } \\frac { ( x - 8 ) ^ { n } } { 9 ^ { n } } \\)\n(a) find the series radius and interval of convergence\n(b) for what values of \\( x \\) does the series converge absolutely?\n(c) for what values of \\( x \\) does the series converge conditionally?\n(a) the interval of convergence is \\( ( - 1,17 ) \\) (simplify your answer type your answer in interval notation )\nthe radius of convergence is

consider the series \\( \\sum _ { n = 0 } ^ { \\infty } \\frac { ( x - 8 ) ^ { n } } { 9 ^ { n } } \\)\n(a) find the series radius and interval of convergence\n(b) for what values of \\( x \\) does the series converge absolutely?\n(c) for what values of \\( x \\) does the series converge conditionally?\n(a) the interval of convergence is \\( ( - 1,17 ) \\) (simplify your answer type your answer in interval notation )\nthe radius of convergence is

Answer

Explanation:

Step1: Recall the formula for the radius of convergence of a geometric series

For a geometric series (\sum_{n = 0}^{\infty}a\cdot r^{n}), it converges when (|r|\lt1). The given series (\sum_{n=0}^{\infty}\frac{(x - 8)^{n}}{9^{n}}=\sum_{n=0}^{\infty}(\frac{x - 8}{9})^{n}) is a geometric series with (a = 1) and (r=\frac{x - 8}{9}).

Step2: Find the radius of convergence

Using the condition for convergence of a geometric series (|r|\lt1). If (r=\frac{x - 8}{9}), then (|\frac{x - 8}{9}|\lt1). The radius of convergence (R) can be found from the inequality (|x - c|\lt R). Comparing (|\frac{x - 8}{9}|\lt1) with (|x - 8|\lt R), we get (R = 9).

Answer:

(9)