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.