5. find the values of x for which the series ∑_{n = 0}^{∞}9(\\frac{x - 8}{9})^{n} converges. a. -1 < x < 17…

Explanation:

Step1: Recognize geometric - series form

The given series $\sum_{n = 0}^{\infty}9(\frac{x - 8}{9})^n$ is a geometric series with the general form $\sum_{n=0}^{\infty}ar^n$, where $a = 9$ and $r=\frac{x - 8}{9}$.

Step2: Apply geometric - series convergence condition

A geometric series $\sum_{n = 0}^{\infty}ar^n$ converges if $|r|\lt1$. So we set up the inequality $\left|\frac{x - 8}{9}\right|\lt1$.

Step3: Solve the absolute - value inequality

We have $- 1\lt\frac{x - 8}{9}\lt1$. Multiply each part of the compound inequality by 9: $-9\lt x - 8\lt9$. Then add 8 to each part: $-9 + 8\lt x-8 + 8\lt9 + 8$, which simplifies to $-1\lt x\lt17$.

Answer:

A. $-1\lt x\lt17$