Find the series' radius of convergence.
-$$\sum _ { n = 0 } ^ { \infty } \frac { ( x - 4 ) ^ { n } } { n ^ { 2 } 6 ^ { n } }$$

Question

Quizplus · Accepted Answer

The radius of convergence $ R $ can be found using the formula for the radius of convergence of a power series: $ R = \frac{1}{\limsup_{n \to \infty} |a_n|^{1/n}} $. Here, $ a_n = \frac{1}{n^2 6^n} $, so $ |a_n|^{1/n} = \frac{1}{n^{2/n} \cdot 6} $. As $ n \to \infty $, $ n^{2/n} \to 1 $, so the limit is $ \frac{1}{6} $. Thus, $ R = 6 $.

Find the Series' Radius of Convergence ∑n=0∞(x−4)nn26n\sum _ { n = 0 } ^ { \infty } \frac { ( x - 4 ) ^ { n } } { n ^ { 2 } 6 ^ { n } }n=0∑∞​n26n(x−4)n​

Find the Series' Radius of Convergence $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 4 ) ^ { n } } { n ^ { 2 } 6 ^ { n } }$