Order of various functions

Question

I am to arrange the following 30 functions in terms of increasing order
( all the lg (log) are to the base 2.
log* represents iterated logarithm.)

Here's my answer (the functions on the same level are of the same order):

1 , $\large n^{1/{\lg n}}$

lglg*n

lg*n,      lg*(lgn)

$\large 2^{\lg*n}$

$\large \ln\ln n$

$\large \sqrt{\lg n}$

$\large \ln n$

$\large \lg^{2} n$

$\large \2^{\sqrt{2\lg n}}$

$\large \sqrt{2}^{\lg n}$

$\large n$ , $\large 2^{\lg n}$

$\large n{\lg n}$ , $\large {\lg (n!)}$

$\large n^{2}$ , $\large 4^{\lg n}$

$\large n^3$

$\large (\lg n)^{\lg n}$ , $\large n^{\lg \lg n}$

$\large 1.5^n$

$\large 2^n$

$\large n2^n$

$\large e^n$

$\large n!$

$\large (n+1)!$

$\large 2^{2^{n}}$

$\large 2^{2^{n+1}}$

I am not sure where to insert (logn)!. I know (logn)! = o((logn)^logn) and logn = o((logn)!), isn't it. But where exactly would it fit. Can anyone help me with that. Also I would really appreciate if someone could verify that my arrangement is right or not.

Thank you.

EDIT : I MEANT TO ASK THE POINT OF INSERTION OF (LOGN)! IN THE ABOVE LIST SORTED IN TERMS OF INCREASING ORDER. Simple.

score 2 · Answer 1 · answered Sep 18 '16 at 16:01

2

Stirling's formula asserts that $$ n! = \Theta(\sqrt{n}(n/e)^n). $$ Thus $$ (\log n)! = \Theta(\sqrt{\log n}((\log n)/e)^{\log n}). $$

answered Sep 18 '16 at 16:01

Yuval Filmus

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Order of various functions

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