By the Inequality of arithmetic and geometric means we have:
1⋅(n−1)≤n/2 or 1⋅(n−1)≤(n/2)2
2⋅(n−2)≤n/2 , or 2⋅(n−2)≤(n/2)2
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if n is even, the last inequality will be n/2≤n/2,
if n is odd, the last inequality will be (n−1)/2⋅(n+1)/2≤(n/2)2
Multiplying all this inequalities (without the square roots), we obtain:
(n−1)!≤(n/2)n−1 and
n!≤2(n/2)n
Therefore, n!nn≥2(n/2)nnn=2n−1
The values 2n−1 are unbounded from above, therefore the sequence nn/n! is unbounded too and hence, it is divergent.
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