Friday, 8 June 2018

limits - Find limlimitsnto+inftyfracsqrt[n]n!n




I tried using Stirling's approximation and d'Alambert's ratio test but can't get the limit. Could someone show how to evaluate this limit?


Answer



Use equivalents:
nn!n(2πn)1nnne=1e(2πn)12n
Now ln(2πn)12n=lnπ+ln2n2nn0, hence
nn!n1e.


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