For what values of x does the series converge absolutely? -$$\sum _ { n = 0 } ^ { \infty } ( 3 x ) ^ { n }$$ A) $x \frac { 1 } { 3 }$ B) $- \frac { 1 } { 3 }

Question

For what values of x does the series converge absolutely?
-$$\sum _ { n = 0 } ^ { \infty } ( 3 x ) ^ { n }$$
A) $x > \frac { 1 } { 3 }$
B) $- \frac { 1 } { 3 } < x < \frac { 1 } { 3 }$
C) $0 < x < \frac { 1 } { 3 }$
D) None

Quizplus · Accepted Answer

The series $\sum _ { n = 0 } ^ { \infty } ( 3 x ) ^ { n }$ is a geometric series with a common ratio of $3x$. For a geometric series to converge absolutely, the absolute value of the common ratio must be less than 1. Therefore, $|3x| < 1$, which simplifies to $-\frac{1}{3} < x < \frac{1}{3}$.

For What Values of X Does the Series Converge Absolutely ∑n=0∞(3x)n\sum _ { n = 0 } ^ { \infty } ( 3 x ) ^ { n }n=0∑∞​(3x)n

For What Values of X Does the Series Converge Absolutely $\sum _ { n = 0 } ^ { \infty } ( 3 x ) ^ { n }$