Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 6 } \right) ^ { n }$
A) $[ - 6,6 )$
B) $( - 6,6 )$
C) $[ - 6,6 ]$
D) $\left( \frac { - 1 } { 6 } , \frac { 1 } { 6 } \right)$
E) $\left[ \frac { - 1 } { 6 } , \frac { 1 } { 6 } \right)$

Question

Quizplus · Accepted Answer

This series is a geometric series with a common ratio of $r = \frac{x}{6}$. The series converges if $|r| < 1$, which means $|\frac{x}{6}| < 1$, or $|x| < 6$. Therefore, the interval of convergence is $(-6, 6)$, without including the endpoints, since at $x = \pm 6$, the series does not converge.

Find the Interval of Convergence of the Power Series ∑n=0∞(x6)n\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 6 } \right) ^ { n }∑n=0∞​(6x​)n

Find the Interval of Convergence of the Power Series $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 6 } \right) ^ { n }$