How to prove $(n+1/2)\log (1+1/n)-1 >0$

Question

I want to show the following statement:

For any integer $n$ with $n\geq 1$, $$\left(n+\frac{1}{2}\right) \log \left(1+\frac{1}{n}\right) -1>0.$$ or equivalently, $$\log \left(n+1\right)-\log \left(n\right) >\frac{2}{2n+1 }. $$ Here log represents the natural log function. I am able to prove $$\log \left(n+1\right)-\log \left(n\right) >\frac{1}{n+1 }. $$ but not much luck with getting $\frac{2}{2n+1 }$. Any help is appreciated.

use that the derivative of $f(x)=\log (x+1)-\log x-\frac{2}{2x+1}$ is negative (it is $-1/(x(x+1)(2x+1)^2)$) on $[1, \infty)$ hence $f$ is decreasing and $\lim_{x \to \infty}f(x)=0$ — Conrad, Jan 08 '22 at 05:19
I believe I can prove that f is decreasing but how do you know that the limit of f is zero? — yi G, Jan 08 '22 at 05:25
$f(x)=\log (1+1/x)-\frac{2}{2x+1}$ and each term goes to zero — Conrad, Jan 08 '22 at 05:25
Related $;-;$ Is $\left(1+\frac1n\right)^{n+1/2}$ decreasing? — dxiv, Jan 08 '22 at 05:32

score 3 · Answer 1 · edited Jan 08 '22 at 05:53

3

For any integer $n$ with $n≥1$,$$\Big(n+\frac{1}{2}\Big)\log\Big(1+\frac{1}{n}\Big)-1>0$$ let $x=\frac{1}{n}$, then $$\Big(n+\frac{1}{2}\Big)\log\Big(1+\frac{1}{n}\Big)-1>0,n\geq1$$ $$\Leftrightarrow (\frac{1}{x}+\frac{1}{2})\log(1+x)-1>0,1\geq x>0$$ $$\Leftrightarrow \log(1+x)>\frac{2x}{x+2},1\geq x>0$$ let$$F(x)=\log(1+x)-\frac{2x}{x+2}$$ then:$$F'(x)=\frac{1}{x+1}-\frac{2(x+2)-2x}{(x+2)^2}=\frac{(x+2)^2-4(x+1)}{(x+1)(x+2)^2}=\frac{x^2}{(x+1)(x+2)^2}>0$$ so $F(x)$ is monotonically increasing function.So $F(x)>F(0)=0$.Done.

edited Jan 08 '22 at 05:53

Gary

31,845

answered Jan 08 '22 at 05:33

anhui-xiaocaigui

242

sorry，is$F'(x)=\frac{1}{1+x}-\frac{2(x+2)-2x}{(x+2)^2}$ – anhui-xiaocaigui Jan 08 '22 at 05:41
You can edit your answer to make correction, no need to make a comment about it. I fixed it for you this time. – Gary Jan 08 '22 at 05:53
haha ,thank you.I just used this editor, not familiar with it, – anhui-xiaocaigui Jan 08 '22 at 05:58
Just below the text of your answer there is a series of options: Share $;$ Cite $;$ Edit $;$ Delete $;$ Flag. Choose Edit and you can edit your answer. $;$ – Gary Jan 08 '22 at 06:00
hahaha……，I get it ,thank you thank you. – anhui-xiaocaigui Jan 08 '22 at 06:01

How to prove $(n+1/2)\log (1+1/n)-1 >0$

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