Find the radius of convergence of the series $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 6 } \right) ^ { n }$ .
A)12
B)14
C)1
D)7
E)6

Question

Quizplus · Accepted Answer

The radius of convergence of the series $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 6 } \right) ^ { n }$ can be found using the formula for the radius of convergence $R$ of a power series $\sum _ { n = 0 } ^ { \infty } a_n (x - c)^n$, which is $R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}}$. In this case, $a_n = \left(\frac{1}{6}\right)^n$, so $\limsup_{n \to \infty} \sqrt[n]{|a_n|} = \limsup_{n \to \infty} \sqrt[n]{\left(\frac{1}{6}\right)^n} = \frac{1}{6}$. Therefore, $R = \frac{1}{1/6} = 6$.

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

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