Identify the interval of convergence of a power series $\sum _ { n = 1 } ^ { \infty } n ( 2 x ) ^ { n }$
A) $- \frac { 1 } { 2 }

Question

Identify the interval of convergence of a power series $\sum _ { n = 1 } ^ { \infty } n ( 2 x ) ^ { n }$
A) $- \frac { 1 } { 2 } < x < 0$
B) $- \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$
C) $0 < x < \frac { 1 } { 2 }$
D) $- \frac { 1 } { 4 } < x < \frac { 1 } { 4 }$
E) $0 < x < \frac { 1 } { 4 }$

Quizplus · Accepted Answer

We use the ratio test to find the interval of convergence: 
$$ \lim_{n \to \infty} \left|\frac{(n+1)(2x)^{n+1}}{n(2x)^n}\right|=\lim_{n \to \infty} \frac{2|x|}{n+1}=0$$
By the ratio test, the series converges for all values where the limit is less than 1, which is equivalent to $-1 < 2x < 1$ or $-\frac{1}{2} < x < \frac{1}{2}$. Therefore, the correct answer is B.

Identify the Interval of Convergence of a Power Series ∑n=1∞n(2x)n\sum _ { n = 1 } ^ { \infty } n ( 2 x ) ^ { n }∑n=1∞​n(2x)n

Identify the Interval of Convergence of a Power Series $\sum _ { n = 1 } ^ { \infty } n ( 2 x ) ^ { n }$