Proving limit with \log(n!) I am trying to calculate the following

osula9a

osula9a

Answered question

2022-01-19

Proving limit with log(n!)
I am trying to calculate the following limits, but I dont

Answer & Explanation

turtletalk75

turtletalk75

Beginner2022-01-19Added 29 answers

You can easily show that 2nn!nn  for  n4 The first inequality is a very standard induction proof, and the second inequality is straight-forward (you're comparing 1×2××n  with  n×n××n).
From there, since f(n)=logn is an increasing function, you have that
nlog(2)log(n!)nlog(n)
This tells you basically everything you will need. For example, for the first one:
limn3nnlognlimn3nlog(n!)limn3nnlog(2)
Andrew Reyes

Andrew Reyes

Beginner2022-01-20Added 24 answers

Stirlings
RizerMix

RizerMix

Expert2022-01-27Added 656 answers

For the first one, use that log(n!)nlog n as n to conclude that limn3nlog(n!)=limn3nnlogn=limn3nlogn=0

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