Find the smallest value of N that will make the inequality hold for all n N.
-$$a _ { n } = \frac { n ! } { n ^ { n } }$$
A) L = 0, N = 7
B) L = ln 4, N = 8
C) L = ln 2, N = 6
D) diverges

Question

Find the smallest value of N that will make the inequality hold for all n > N.
-$$a _ { n } = \frac { n ! } { n ^ { n } }$$
A) L = 0, N = 7
B) L = ln 4, N = 8
C) L = ln 2, N = 6
D) diverges

Quizplus · Accepted Answer

The sequence $a_n = \frac{n!}{n^n}$ tends to 0 as $n$ approaches infinity. This is because the factorial $n!$ grows slower than the exponential function $n^n$, making the fraction smaller and smaller as $n$ increases. There is no specific smallest $N$ that universally makes the inequality hold for all $n > N$, but for practical purposes in this context, choosing $N = 7$ as a point where the trend towards 0 becomes clear is reasonable. The sequence does not diverge; instead, it converges to 0, eliminating the other options.

Find the Smallest Value of N That Will Make the Inequality