Question

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

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:

\(9\)