Determine if the sequence is monotonic and if it is bounded.
-$$a _ { n } = \frac { 3 ^ { n } } { ( 3 n ) ! }$$
A) nondecreasing; bounded
B) decreasing; bounded
C) increasing; unbounded
D) not monotonic; bounded

Question

Quizplus · Accepted Answer

The sequence $a_n = \frac{3^n}{(3n)!}$ is decreasing because as $n$ increases, the denominator increases much faster than the numerator due to the factorial, making each subsequent term smaller than the previous one. It is bounded because the sequence is always positive (since both the numerator and denominator are positive for all $n$) and decreases towards 0, meaning it has a lower bound of 0 and an upper bound determined by the first term of the sequence.

Determine If the Sequence Is Monotonic and If It Is an=3n(3n)!a _ { n } = \frac { 3 ^ { n } } { ( 3 n ) ! }an​=(3n)!3n​

Determine If the Sequence Is Monotonic and If It Is $a _ { n } = \frac { 3 ^ { n } } { ( 3 n ) ! }$