Find the series' radius of convergence.
-$$\sum _ { n = 0 } ^ { \infty } \frac { n ( n + 1 ) ( n + 2 ) } { 8 ^ { n } } ( x - \pi ) ^ { n }$$

Question

Quizplus · Accepted Answer

The radius of convergence $R$ can be found using the ratio test, which involves taking the limit as $n$ approaches infinity of the absolute value of the ratio of the $(n+1)$th term to the $n$th term of the series. For the given series, the ratio of the $(n+1)$th term to the $n$th term simplifies to $\frac{|x-\pi|}{8}$ as $n$ approaches infinity. Setting this less than 1 for convergence, we find $|x-\pi| < 8$, indicating that the radius of convergence is $R = 8$.

Find the Series' Radius of Convergence ∑n=0∞n(n+1)(n+2)8n(x−π)n\sum _ { n = 0 } ^ { \infty } \frac { n ( n + 1 ) ( n + 2 ) } { 8 ^ { n } } ( x - \pi ) ^ { n }n=0∑∞​8nn(n+1)(n+2)​(x−π)n

Find the Series' Radius of Convergence $\sum _ { n = 0 } ^ { \infty } \frac { n ( n + 1 ) ( n + 2 ) } { 8 ^ { n } } ( x - \pi ) ^ { n }$