Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. $$\sum _ { n = 0 } ^ { \infty } 10 \left( \frac { x - 8 } { 10 } \right) ^ { n }$$
A) $- \frac { 100 } { 18 - x } , - 2

Question

Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. $$\sum _ { n = 0 } ^ { \infty } 10 \left( \frac { x - 8 } { 10 } \right) ^ { n }$$
A) $- \frac { 100 } { 18 - x } , - 2 < x < 18$
B) $\frac { 100 } { 18 - x } , - 2 < x < 18$
C) $\frac { 100 } { 28 + x } , - 12 < x < 28$ 
D) $\frac { 100 } { 28 - x } , - 12 < x < 28$
E) The series diverges for all $x$.

Quizplus · Accepted Answer

This series is a geometric series with the first term $a = 10$ and common ratio $r = \frac{x - 8}{10}$. A geometric series converges if and only if $|r| < 1$. Thus, $\left|\frac{x - 8}{10}\right| < 1$, which simplifies to $-1 < \frac{x - 8}{10} < 1$, leading to $-2 < x - 8 < 10$, and further to $-2 < x < 18$. The sum of a converging geometric series is given by $S = \frac{a}{1 - r}$, which in this case is $S = \frac{10}{1 - \frac{x - 8}{10}} = \frac{10}{\frac{10 - x + 8}{10}} = \frac{100}{18 - x}$.

Find All Values of X for Which the Series Converges ∑n=0∞10(x−810)n\sum _ { n = 0 } ^ { \infty } 10 \left( \frac { x - 8 } { 10 } \right) ^ { n }n=0∑∞​10(10x−8​)n

Find All Values of X for Which the Series Converges $\sum _ { n = 0 } ^ { \infty } 10 \left( \frac { x - 8 } { 10 } \right) ^ { n }$