Find all values of $x$ for which the series
$\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { x - 8 } { 9 } \right) ^ { n }$ converges.
A) $- 1

Question

Find all values of $x$ for which the series
$\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { x - 8 } { 9 } \right) ^ { n }$ converges. 
A) $- 1 < x < 17$
B) $- 10 < x < 18$
C) $- 1 < x < 18$
D) $- 18 < x < 19$
E) $- 9 < x < 19$

Quizplus · Accepted Answer

The series is a geometric series with the common ratio $r = \frac{x - 8}{9}$. For a geometric series to converge, $|r| < 1$. Thus, $\left|\frac{x - 8}{9}\right| < 1$, which simplifies to $-1 < \frac{x - 8}{9} < 1$. Solving these inequalities gives $-1 < x - 8 < 9$, which further simplifies to $-1 + 8 < x < 9 + 8$, leading to $7 < x < 17$. Therefore, the correct interval for $x$ is $-1 < x < 17$.

Find All Values Of xxx For Which the Series ∑n=0∞9(x−89)n\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { x - 8 } { 9 } \right) ^ { n }∑n=0∞​9(9x−8​)n

Find All Values Of $x$ For Which the Series $\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { x - 8 } { 9 } \right) ^ { n }$