How to compute the asymptotic growth of ((n),(\log n))?

nrgiizr0ib6

nrgiizr0ib6

Answered question

2022-04-21

How to compute the asymptotic growth of (nlogn)?

Answer & Explanation

Elsaidrge

Elsaidrge

Beginner2022-04-22Added 11 answers

You can make use of Stirling's approximation.
logn!=nlognn+12log2πn+O(1n)
log(nlogn)=logn!log((nlogn)!)log((logn)!)
=nlogn(nlogn)log(nlogn)+O(lognloglogn)
=nlogn(nlogn)(logn+log(1lognn))+O(lognloglogn)
=log2n+O(lognloglogn)
Thus your binomial coefficient is
nlogn+O(loglogn)
We can make it more accurate by computing the leading terms in
O(lognloglogn)

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