Find the sum of the series as a function of x.
-$$\sum _ { n = 1 } ^ { \infty } \left( \frac { x - 6 } { 3 } \right) ^ { n }$$
A) $- \frac { x - 6 } { x - 9 }$
B) $- \frac { x - 6 } { x + 9 }$
C) $- \frac { x - 6 } { x + 3 }$
D) $- \frac { x - 6 } { x - 15 }$

Question

Quizplus · Accepted Answer

This series is a geometric series with the first term $a = \frac{x-6}{3}$ and common ratio $r = \frac{x-6}{3}$. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, provided $|r| < 1$. Substituting $a$ and $r$ into this formula gives $S = \frac{\frac{x-6}{3}}{1 - \frac{x-6}{3}} = \frac{x-6}{3 - (x-6)} = \frac{x-6}{9-x}$, which simplifies to $-\frac{x-6}{x-9}$.

Find the Sum of the Series as a Function of X