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oleifere45

oleifere45

Answered question

2022-06-22

Prove n log ( 1 + 1 n ) log ( 1 + 1 n + 1 ) < log ( 1 + 1 n + 1 )
I'm trying to prove the above inequality, assuming n 1. I've been working on this one using log properties and trying to reduce this inequalitiy to simpler ones. Though!, is it even correct? or am I trying to prove a falsity? Thanks, I strongly suspect I'm wasting my time on this one??

Answer & Explanation

iceniessyoy

iceniessyoy

Beginner2022-06-23Added 27 answers

Notice that n log ( 1 + 1 n ) log ( 1 + 1 n + 1 ) < log ( 1 + 1 n + 1 ) can be rewritten as
n log ( 1 + 1 n ) 2 log ( 1 + 1 n + 1 ) < 0. Since 1 + 1 n + 1 < 1 + 1 n and log is an increasing function, we have that
n log ( 1 + 1 n ) 2 log ( 1 + 1 n + 1 ) > ( n 2 ) log ( 1 + 1 n + 1 ) .
But, for n 2, the right hand side is greater than zero, so
n log ( 1 + 1 n ) 2 log ( 1 + 1 n + 1 ) > 0 n log ( 1 + 1 n ) log ( 1 + 1 n + 1 ) > log ( 1 + 1 n + 1 ) .

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