Determine if the sequence is monotonic and if it is bounded.
-$$\frac { \mathrm { n } ! } { 5 ^ { n } }$$
A) nondecreasing; unbounded
B) nondecreasing; bounded
C) not monotonic; unbounded
D) nonincreasing; unbounded

Question

Quizplus · Accepted Answer

The sequence is nondecreasing because $$\frac { ( n + 1 ) ! } { 5 ^ { n + 1 } } = \frac { ( n + 1 ) n ! } { 5 ^ { n + 1 } } = \frac { n ! } { 5 ^ { n } } \cdot \frac { n + 1 } { 5 } > \frac { n ! } { 5 ^ { n } }$$ for all $n$. The sequence is unbounded because $$\lim _ { n \rightarrow \infty } \frac { n ! } { 5 ^ { n } } = \lim _ { n \rightarrow \infty } \frac { n ! } { 5 ^ { n } } \cdot \frac { 5 ^ { n } } { n ! } = \lim _ { n \rightarrow \infty } 5 ^ { n } = \infty$$

Determine If the Sequence Is Monotonic and If It Is n!5n\frac { \mathrm { n } ! } { 5 ^ { n } }5nn!​

Determine If the Sequence Is Monotonic and If It Is $\frac { \mathrm { n } ! } { 5 ^ { n } }$