I have the following function $a_n=$ $\frac{\ln{n}}{n}$ and I want to find its decreasing interval from (0 $,$ $\infty$). I tried two methods and got different results.
if I take the derivative i will get $\frac{1-\ln{n}}{n^2}$. So the decreasing interval is from (1,$\infty$).
But if I take use the following
$a_{n+1} \leq a_n \implies \frac{\ln{n+1}}{n+1} \leq \frac{\ln{n}}{n} $ which is true for $n\geq3$.
How am I getting different values? is there a mistake with the second method ?