Find the radius of convergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } ( x + 1 ) ^ { n } } { n ^ { n } }$ .
A)0
B)2
C)3
D)4
E) $\frac { 1 } { 2 }$
F)1
G)5
H) $\infty$

Question

Find the radius of convergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } ( x + 1 ) ^ { n } } { n ^ { n } }$ .
A)0
B)2
C)3
D)4
E) $\frac { 1 } { 2 }$ 
F)1
G)5
H) $\infty$

Quizplus · Accepted Answer

The radius of convergence of the series can be found using the root test, which involves taking the $n$th root of the absolute value of the $n$th term and then taking the limit as $n$ approaches infinity. For the given series, the $n$th term is $\frac{3^n(x+1)^n}{n^n}$. Taking the $n$th root of the absolute value of this term gives $\frac{3|x+1|}{n}$. As $n$ approaches infinity, this expression approaches 0 for any finite value of $x$, indicating that the series converges for all $x$. Therefore, the radius of convergence is $\infty$.

Find the Radius of Convergence of the Series ∑n=1∞3n(x+1)nnn\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } ( x + 1 ) ^ { n } } { n ^ { n } }∑n=1∞​nn3n(x+1)n​

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