# Proving inequality

• Mar 19th 2013, 07:02 PM
Kanwar245
Proving inequality
Prove that for all sufficiently large n,

$\frac{(1+\log2)n}{\log\frac{n}{2}} \leq \frac{2n}{\log n}$
• Mar 19th 2013, 09:29 PM
Prove It
Re: Proving inequality
Well for starters, find the smallest value where the inequality holds true, then show the right hand function grows faster than the left hand function.
• Mar 19th 2013, 09:29 PM
princeps
Re: Proving inequality
First , find by guessing some small value for which inequality holds , then prove inequality using math induction .
• Mar 19th 2013, 09:30 PM
chiro
Re: Proving inequality
Hey Kanwar245.

Hint: Try taking log(n/2) to the RHS and use the fact that log(n/2) = log(n) - log(2).
• Mar 19th 2013, 09:31 PM
Prove It
Re: Proving inequality
Quote:

Originally Posted by princeps
First , find by guessing some small value for which inequality holds , then prove inequality using math induction .

Is mathematical induction wise here? We aren't told that n only takes on integer values...
• Mar 19th 2013, 09:38 PM
princeps
Re: Proving inequality
Quote:

Originally Posted by Prove It
Is mathematical induction wise here? We aren't told that n only takes on integer values...

Common notation for integer variable is a letter "n" .
• Mar 20th 2013, 12:32 PM
Kanwar245
Re: Proving inequality
Quote:

Originally Posted by Prove It
Well for starters, find the smallest value where the inequality holds true, then show the right hand function grows faster than the left hand function.

How can I show that the function on the right side grows faster than the function on the left side?
• Mar 20th 2013, 04:30 PM
Prove It
Re: Proving inequality
Quote:

Originally Posted by Kanwar245
How can I show that the function on the right side grows faster than the function on the left side?

Take the derivative of each side. Show the right hand derivative is greater than the left.