Simple inequality help I need a function f(x) that satisfies the properties bellow for all

Jakayla Benton

Jakayla Benton

Answered question

2022-05-01

Simple inequality help
I need a function f(x) that satisfies the properties bellow for all integers k
logk+1k+1-log1+1k+fk+1-fk<0
limkf(k)=0
I don't think it should be very hard sense if I let f(x)=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.

Answer & Explanation

Payton Cantrell

Payton Cantrell

Beginner2022-05-02Added 15 answers

The f you seek cannot exist.
Define f(k+1)f(k)εk+1
f(k)=f(1)j=2kεj,
if we assume that
εk+1>log(k+1)k+1log(1+1k)logkk
for k large enough, we would necessarily have f(k) as k by the comparison test.

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