Big O boundary condition truth value

Note: logs below are base 2. (Not sure how to do subscripts here)

Wondering if the below equation is true when thinking asymptotically (Computer Science)

$log_2((n!)^n) = O(n \sin(n \frac{\pi}{2}) + \log_2{n})$

But I'm not sure how to compute this.

I'm guessing we need to take the log of both sides of the following equation:

$log_2((n!)^n) < n sin(n (pi/2)) + log_2(n)$

getting us:

$log_2(log_2((n!)^n)) < log_2 (n sin(n (pi/2)) + log_2(n) )$

Not sure where to go from there.

• You should know that all log have the same big O limit, as they differ by a constant. Secondly, an underscore (within math mode, that is $signs) will get you to subscript. Sep 8 '10 at 0:57 • Also, I'd try and use Stirling's approximation for$n!$and the fact that$log(n!^n) = nlog(n!)$, and try to work it from there. Or something like that. Sep 8 '10 at 1:00 • Thanks, I tried to make it look nicer with your suggestions...hope it helps the readability. – Sev Sep 8 '10 at 1:04 1 Answer If you are interested in computer science, a very useful thing to remember is that $$\log n! = \theta(n \log n)$$ Now given this, can you tell what$f(n)$is, if$ \log (n!^{n}) = \theta(f(n))$? How would that compare to the right hand side? • Theta(n log(n!)) -- but in my equation we need to take the log of that again, which gives us log(n log(n!))...any tips on where to go from there? – Sev Sep 8 '10 at 2:46 • @Sev: Why do you need to take log again? what is$\log (a^n)\$? Sep 8 '10 at 2:50
• because originally it was log (log((n!)^n)) -- now i see that log(n!^n) is n log n! but what about the outer log?
– Sev
Sep 8 '10 at 2:53
• @Sev: If log (n!) ~ nlogn, what is nlog(n!)? What is the log of that? Sep 8 '10 at 2:55
• @Moron: nlog(n!) must be n* nlogn which is n^2 log n?
– Sev
Sep 8 '10 at 2:58