How do you apply the ratio test to determine if $Sigma 1/sqrt(n!)$ from $n=[1,oo)$ is convergent to divergent?

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Answer 1 · 2017-03-09T01:23:15

The series:

$sum_(n=1)^oo 1/sqrt(n!)$

is convergent.

Evaluate the ratio:

$abs(a_(n+1)/a_n) = abs ((1/sqrt((n+1)!))/(1/sqrt(n!))) = sqrt((n!)/((n+1)!)) = 1/sqrt(n+1)$

We have that:

$lim_(n->oo) abs(a_(n+1)/a_n) = lim_(n->oo) 1/sqrt(n+1) = 0$

so the series is convergent.

How do you apply the ratio test to determine if Sigma 1/sqrt(n!)Sigma 1/sqrt(n!) from n=[1,oo)n=[1,oo) is convergent to divergent?