Determine if the sequence is monotonic and if it is bounded.
-$$a _ { n } = \frac { 4 ^ { n } } { ( 4 n ) ! }$$
A) not monotonic; bounded
B) nondecreasing; bounded
C) increasing; unbounded
D) decreasing; bounded

Question

Quizplus · Accepted Answer

To determine if the sequence is monotonic, we take the first derivative of the sequence:
$$a_{n+1}'-a_n'=\frac{4^{n+1}\cdot(4n-3)!-4^n\cdot(4n+4)!}{(4n+4)!^2\cdot(4n)!^2}=\frac{4^n(4n-3)!(4n+1)}{(4n+4)!^2\cdot(4n)!}<0$$
Therefore, the sequence is decreasing. To determine if the sequence is bounded, we can use the ratio test:
$$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}=\lim_{n\rightarrow\infty}\frac{4}{(4n+4)(4n+3)(4n+2)(4n+1)}=0$$
By the ratio test, the series converges to $0$, which means the sequence is bounded. Therefore, the answer is D.

Determine If the Sequence Is Monotonic and If It Is an=4n(4n)!a _ { n } = \frac { 4 ^ { n } } { ( 4 n ) ! }an​=(4n)!4n​

Determine If the Sequence Is Monotonic and If It Is $a _ { n } = \frac { 4 ^ { n } } { ( 4 n ) ! }$