Use the Root Test to determine if the series converges or diverges.
-$$\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 3 n } } { [ ( 3 n ) ! ] ^ { n } }$$
A) Converges
B) Diverges

Question

Quizplus · Accepted Answer

The Root Test involves taking the nth root of the absolute value of the nth term and examining the limit as n approaches infinity. For the given series, we apply the Root Test as follows: $$ \lim _ { n \rightarrow \infty } \sqrt[n]{ \left| \frac { ( n ! ) ^ { 3 n } } { [ ( 3 n ) ! ] ^ { n } } \right| } $$ Using properties of factorials and Stirling's approximation or direct comparison, we can show that this limit approaches a value less than 1, which implies convergence according to the Root Test.

Use the Root Test to Determine If the Series Converges ∑n=1∞(n!)3n[(3n)!]n\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 3 n } } { [ ( 3 n ) ! ] ^ { n } }n=1∑∞​[(3n)!]n(n!)3n​

Use the Root Test to Determine If the Series Converges $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 3 n } } { [ ( 3 n ) ! ] ^ { n } }$