Find the radius of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 5 ^ { n } }$
A) $\frac { 1 } { 25 }$
B) $\frac { 1 } { 5 }$
C) 5
D) $\infty$
E) 25

Question

Quizplus · Accepted Answer

The radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 5 ^ { n } }$ can be found using the ratio test, which in this context simplifies to finding the limit of the absolute value of the ratio of consecutive terms. This gives us $|x/5|$, and for convergence, we need $|x/5| < 1$, which simplifies to $|x| < 5$. Therefore, the radius of convergence is 5.

Find the Radius of Convergence of the Power Series ∑n=0∞(−1)nxn5n\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 5 ^ { n } }∑n=0∞​5n(−1)nxn​

Find the Radius of Convergence of the Power Series $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 5 ^ { n } }$