How do you proof lim x → ∞ (n^(x+1)) / ((x+1)!) = 0 ?

Question

#(n^(x+1))/((x+1)!)# *

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Answer 1 · 2018-04-11T16:33:01

below

Assuming the Stirling approx is allowed, ie: # ln Q! approx Q ln (Q) - Q#

#lim_(x to oo) (n^(x+1))/((x+1)!)#

let #z = x+1#

#= lim_(z to oo) exp( ln( (n^z)/(z!)))#

#= lim_(z to oo) exp( z ln n - ln (z!))#

#= lim_(z to oo) exp( z ln n - z ln z + z)#

#= lim_(z to oo) exp( z ln n - z ln z + z ln e)#

#= lim_(z to oo) exp( ln ((n e)/z)^z )#

#= lim_(z to oo) ((n e)/z)^z #

For finite #n#, # lim_(z to oo) ((n e)/z) = 0 #, and so # lim_(z to oo) ((n e)/z)^z = 0#

Sure there is a better way to express that last sentence.