Find the values of $x$ for which the series $\sum _ { n = 0 } ^ { \infty } 5 \left( \frac { x } { 4 } \right) ^ { n }$ converges.
A) $- 5

Question

Find the values of $x$ for which the series $\sum _ { n = 0 } ^ { \infty } 5 \left( \frac { x } { 4 } \right) ^ { n }$ converges. 
A) $- 5 < x < 0$
B) $0 < x < 5$
C) $- 4 < x < 4$
D) $- 5 < x < 5$
E) $0 < x < 4$

Quizplus · Accepted Answer

The series is a geometric series with the common ratio $r = \frac{x}{4}$. For a geometric series to converge, the absolute value of the common ratio must be less than 1, i.e., $|r| < 1$. Therefore, $|\frac{x}{4}| < 1$, which simplifies to $-1 < \frac{x}{4} < 1$, and further to $-4 < x < 4$.

Find the Values Of xxx For Which the Series ∑n=0∞5(x4)n\sum _ { n = 0 } ^ { \infty } 5 \left( \frac { x } { 4 } \right) ^ { n }∑n=0∞​5(4x​)n

Find the Values Of $x$ For Which the Series $\sum _ { n = 0 } ^ { \infty } 5 \left( \frac { x } { 4 } \right) ^ { n }$