Find the radius of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { 2 \cdot 5 \cdot 8 \cdots ( 3 n - 1 ) } { 3 \cdot 7 \cdot 11 \cdots ( 4 n - 1 ) } ( x + 1 ) ^ { n }$ .
A)0
B) $\frac { 4 } { 3 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 3 } { 2 }$
E) $\frac { 1 } { 2 }$
F)2

G) $\frac { 2 } { 3 }$
H) $\infty$

Question

Find the radius of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { 2 \cdot 5 \cdot 8 \cdots ( 3 n - 1 ) } { 3 \cdot 7 \cdot 11 \cdots ( 4 n - 1 ) } ( x + 1 ) ^ { n }$ .
A)0
B) $\frac { 4 } { 3 }$ 
C) $\frac { 3 } { 4 }$ 
D) $\frac { 3 } { 2 }$ 
E) $\frac { 1 } { 2 }$ 
F)2

G) $\frac { 2 } { 3 }$ 
H) $\infty$

Quizplus · Accepted Answer

The ratio test is used to find the radius of convergence. The ratio of consecutive terms simplifies to $\frac{3n}{4n}$, which approaches $\frac{3}{4}$ as $n$ approaches infinity. Thus, the radius of convergence is $\frac{4}{3}$.

Find the Radius of Convergence Of ∑n=1∞2⋅5⋅8⋯(3n−1)3⋅7⋅11⋯(4n−1)(x+1)n\sum _ { n = 1 } ^ { \infty } \frac { 2 \cdot 5 \cdot 8 \cdots ( 3 n - 1 ) } { 3 \cdot 7 \cdot 11 \cdots ( 4 n - 1 ) } ( x + 1 ) ^ { n }∑n=1∞​3⋅7⋅11⋯(4n−1)2⋅5⋅8⋯(3n−1)​(x+1)n

Find the Radius of Convergence Of $\sum _ { n = 1 } ^ { \infty } \frac { 2 \cdot 5 \cdot 8 \cdots ( 3 n - 1 ) } { 3 \cdot 7 \cdot 11 \cdots ( 4 n - 1 ) } ( x + 1 ) ^ { n }$