Find the radius of convergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 3 n + 7 } } { 3 n + 7 }$ .
A)1
B)3
C)10
D)4
E)6

Question

Quizplus · Accepted Answer

The radius of convergence of the series can be found using the ratio test, which involves taking the limit of the absolute value of the ratio of consecutive terms. For the given series, the general term is $a_n = \frac { ( - 1 ) ^ { n } x ^ { 3 n + 7 } } { 3 n + 7 }$. The ratio of the (n+1)th term to the nth term, in absolute value, simplifies to $|x|^3$. Setting this less than 1 for convergence, $|x|^3 < 1$, implies $|x| < 1$. Therefore, the radius of convergence is 1.

Find the Radius of Convergence of the Series ∑n=1∞(−1)nx3n+73n+7\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 3 n + 7 } } { 3 n + 7 }∑n=1∞​3n+7(−1)nx3n+7​

Find the Radius of Convergence of the Series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 3 n + 7 } } { 3 n + 7 }$