# How can I prove that $\ln(n)/n\ge\ln(n+1)/(n+1)$ for $n\ge3$ [closed]

I need to prove that $$a_n\ge a_{n+1}$$, where $$a_n=\frac{\ln\left(n\right)}{n}$$ How can I do it?

• This is true for $n\geq 3$. – bjorn93 Jan 10 at 16:02

If $$f(x) = \frac{{\ln (x)}} {x}$$ then $$f'(x) = \frac{{1 - \ln (x)}} {{x^2 }}$$ This means that $$f$$ is decreasing and this prove what you want. which is negative for $$x \geq e$$ .
If you differentiate $$\frac{\ln x}{x}$$, you will see that it is decreasing for $$x> e$$.