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1/ln((2)!) + 1/ln((3)!) +1 /ln((4)!) +...+ 1/ln((n)!)

the first thing i thought of was de lambert

lim An+1/An

n->inf

=ln(x!)/ln((x+1)!)

=ln(x!)/ln(x!*(x+1))

=ln(x!)/[ln(x!)+ln(x+1)]

=1-ln(x+1)/[ln(x!)+ln(x+1)]=1

which doesn't tell me anything about the convergence /divegence of the series.

have i done something wrong somewhere here, or is there another way to solve this?