Jakayla Benton

2022-05-01

Simple inequality help

I need a function $f\left(x\right)$ that satisfies the properties bellow for all integers $k$

$\frac{\mathrm{log}\left(k+1\right)}{k+1}-\mathrm{log}\left(1+\frac{1}{k}\right)+f\left(k+1\right)-f\left(k\right)<0$

$\underset{k\Rightarrow \mathrm{\infty}}{lim}f\left(k\right)=0$

I don't think it should be very hard sense if I let $f\left(x\right)=0$, the entire thing is already very close to zero. In addition if you find a function that doesn't work for the first few values 1,2,3.. etc, thats fine too. I would appreciate any help, thanks.

Payton Cantrell

Beginner2022-05-02Added 15 answers

The f you seek cannot exist.

Define$f(k+1)-f\left(k\right)\mathrm{\u25b3}-{\epsilon}_{k+1}$

$f\left(k\right)=f\left(1\right)-\sum _{j=2}^{k}{\epsilon}_{j},$

if we assume that

$\epsilon}_{k+1}>\frac{\mathrm{log}(k+1)}{k+1}-\mathrm{log}(1+\frac{1}{k})\sim \frac{\mathrm{log}k}{k$

for k large enough, we would necessarily have$f\left(k\right)\to -\mathrm{\infty}$ as $k\to \mathrm{\infty}$ by the comparison test.

Define

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