Answer to Question #161765 in Calculus for Vishal

Question #161765

Prove that the series ∞n=1 n^n/n! diverges


1
Expert's answer
2021-02-24T06:39:43-0500

By the Inequality of arithmetic and geometric means we have:

1(n1)n/2\sqrt{1\cdot (n-1)}\leq n/2 or 1(n1)(n/2)21\cdot (n-1)\leq (n/2)^2

2(n2)n/2\sqrt{2\cdot (n-2)}\leq n/2 , or 2(n2)(n/2)22\cdot (n-2)\leq (n/2)^2

...

if n is even, the last inequality will be n/2n/2n/2\leq n/2,

if n is odd, the last inequality will be (n1)/2(n+1)/2(n/2)2(n-1)/2\cdot (n+1)/2\leq (n/2)^2

Multiplying all this inequalities (without the square roots), we obtain:

(n1)!(n/2)n1(n-1)!\leq (n/2)^{n-1} and

n!2(n/2)nn!\leq 2(n/2)^n

Therefore, nnn!nn2(n/2)n=2n1\frac{n^n}{n!}\geq\frac{n^n}{2(n/2)^n}=2^{n-1}

The values 2n12^{n-1} are unbounded from above, therefore the sequence nn/n!{n^n}/n! is unbounded too and hence, it is divergent.



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