Algorithm - How to show Big O for 2 functions

Question

I believe that Ln(n) is <= 2^(sqrt(2*lg(n))), so I am trying to prove it

I know that you can show Big O by showing Little O, so I am trying to take the limit as n approaches infinity of Ln(n)/2^(sqrt(2*lg(n)))

However I am finding it hard to prove this using the limit definition

I would appreciate any help to see how to solve this! Thanks

@Kneelac: You realize those are only a constant-factor from the other, which is ignored in big-O, right? — Mooing Duck, Mar 16 '17 at 01:57
@Kneelac: Did you mean ln on the left and lg on the right? — Mooing Duck, Mar 16 '17 at 01:42
@Kneelac Right now you have 2 ^ (2*lg(n)) ^ (1/2). Is that the same as sqrt(2 ^ (2*lg(n)))? Or 2^(sqrt(2*lg(n)))? — Mooing Duck, Mar 16 '17 at 01:33
Please make your formulas uniquely parseable. What is lgn? Is 2lgn the same as 2*log(n)? — , Mar 16 '17 at 01:20
2 ^ (2lg(n)) ^ (1/2) is the same as 2^(sqrt(2lg(n))) Mooing Duck — Kneelac, Mar 16 '17 at 01:36
@MooingDuck Nope, should be ln on the left and lg on the right — Kneelac, Mar 16 '17 at 01:45
Sorry about that, it should be slightly better to read. Yeah, 2lgn is the same as 2*log(n) — Kneelac, Mar 16 '17 at 01:24
I don't understand what you're asking. The title says "big O", then you change your mind and it's an inequality, then it's suddenly big O again. Could you clarify what is the problem? — quicksort, Mar 17 '17 at 19:41

score 0 · Answer 1 · answered Mar 16 '17 at 09:48

Interpreting lg(n) as log2(n), set n=2^x so that the inequality transforms to

ln(2)*x <= 2^sqrt(2*x)

Now apply the square root,

sqrt(2*ln(2)) * sqrt(x/2) <= 2^sqrt(x/2)

and we are left to prove that for y>0

2^y/y > sqrt(2*ln(2))

Indeed, the minimum of the left side can be found for 0=ln(2)*y-1 or y=1/ln(2) with minimum value 2^(1/ln(2))*ln(2). Numerically, the gap between 1.88416938536372 and 1.1774100225154747 is large enough to exclude any floating point errors. Thus all the equivalent inequalities including the original one are true.

Algorithm - How to show Big O for 2 functions

1 Answers1