Inequality about logarithm

Question

I have tried to prove the following inequality:

$$ \left(1+\frac{\log n}{n}\right)^n \gt\frac{n+1}{2}, \mbox{for}\;n\in\{2,3,\ldots\} $$

which seems to be correct (confirmed by numerical result).

Can anyone give me some help or hint? Thanks a lot.

Answer 1

I just want to give a hint: Since the inequality is claimed to be true for all natural numbers greater than one, I would try a proof by induction. It consists of four steps:

1. You show that the inequality is true for $n = 2$.
2. You assume that th inequality is true for an $n$.
3. You show that the inequality is true for $n + 1$.
4. You conclude that the inequality was shown to be true for all $n \geq 2$.

Answer 2

$$ \left(1+\frac{\log n}{n}\right)^n\gt\frac{n+1}{2} $$ Let $n=2$, then $$ \left(1+\frac{\log 2}{2}\right)^2\gt\frac{2+1}{2} $$ $$ 1+\log 2+\frac14(\log 2)^2\gt1+\frac12 $$ Let $n=k+1$, then $$\left(1+\frac{\log(k+1)}{k+1}\right)^{k+1}\gt\left(1+\frac{\log k}{k}\right)^k+\frac12$$ Applying the induction hypothesis, we have $$\left(1+\frac{\log k}{k}\right)^k+\frac12\gt\frac{k+1}{2}+\frac12$$ Therefore $$\left(1+\frac{\log(k+1)}{k+1}\right)^{k+1}\gt\frac{(k+1)+1}{2}$$ Which proves the hypothesis for $n\geq 2$.