By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit $L$ and the smallest integer $N$ such that $\left| a _ { n } - L \right| < 0.01$ for $n \geq N$ ; otherwise indicate divergence.
- $a _ { n } = \frac { n ! } { n ^ { n } }$

Question

By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit $L$ and the smallest integer $N$ such that $\left| a _ { n } - L \right| < 0.01$ for $n \geq N$; otherwise indicate divergence.
-$$a _ { n } = \frac { n ! } { n ^ { n } }$$
A) L = ln 2, N = 6
B) L = 0, N = 7
C) L = ln 4, N = 8
D) diverges

Quizplus · Accepted Answer

The series converges to 0 because the factorial $n!$ grows slower than $n^n$, making $a_n$ approach 0 as $n$ increases. To find the smallest $N$ such that $\left| a _ { n } - L \right| < 0.01$, one would need to calculate the terms of the sequence until the difference is less than 0.01, which happens at $N = 7$.

By Calculating an Appropriate Number of Terms, Determine If the Series