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 9
(b) the series converges absolutely on the interval

Explanation:

Step1: Recall the concept of absolute convergence

A power series \(\sum_{n = 0}^{\infty}a_{n}(x - c)^{n}\) converges absolutely if \(\sum_{n=0}^{\infty}\vert a_{n}(x - c)^{n}\vert\) converges. For the series \(\sum_{n = 0}^{\infty}\frac{(x - 8)^{n}}{9^{n}}\), we can rewrite it as \(\sum_{n=0}^{\infty}(\frac{x - 8}{9})^{n}\), which is a geometric series of the form \(\sum_{n = 0}^{\infty}r^{n}\) with \(r=\frac{x - 8}{9}\).

Step2: Use the convergence condition for geometric series

A geometric series \(\sum_{n=0}^{\infty}r^{n}\) converges if \(\vert r\vert<1\). For our series, we have \(\vert\frac{x - 8}{9}\vert<1\).

Step3: Solve the inequality

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Answer:

The series converges absolutely on the interval \((-1,17)\)