How do you use the ratio test to test the convergence of the series $∑(n!)/(n^n)$ from n=1 to infinity?

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Answer 1 · 2018-08-05T14:38:49

The series converges.

The term of the series is

$a_n=(n!)/(n^n)$

The ratio test is

$|a_(n+1)/a_n|=|((n+1)!)/((n+1)^(n+1))*n^n/(n!)|$

$=|((n+1))/((n+1)^(n+1))*n^n|$

$=|(n^n)/(n+1)^n|$

Therefore,

$lim_(n->oo)|a_(n+1)/a_n|=lim_(n->oo)|(n^n)/(n+1)^n|$

$=lim_(n->oo)(n/(n+1))^n$

$=lim_(n->oo)((n+1)/(n))^-n$

$=1/e$

As

$lim_(n->oo)|a_(n+1)/a_n| <1$

The series converges.

How do you use the ratio test to test the convergence of the series ∑(n!)/(n^n)∑(n!)/(n^n) from n=1 to infinity?