（Ⅰ）求函数f（x）的单调区间，并判断是否有极值；

（Ⅱ）若对任意的x＞1，恒有ln（x﹣1）+k+1≤kx成立，求k的取值范围；

（Ⅲ）证明： $\frac{\ln 2}{2^2}+\frac{\ln 3}{3^2}+\cdots +\frac{\ln n}{n^2}<\frac{2n^2-n-1}{4\left(n+1\right)}$ （n∈N₊，n≥2）．

    Zhang Lin is a middle school student. He is a good boy. Uncle Wu lives next to him. Uncle Wu has no child and can't see anything. He works in the factory (工厂) near Zhang Lin's school. He goes to work at 7:30 in the morning and comes home at 4:30 in the afternoon. Zhang Lin walks to school at 8:00 in the morning and comes home at the same time as Uncle Wu in the afternoon. On weekdays Zhang Lin gets up early to take Uncle Wu to the factory. After school he takes him home. On Sundays Zhang Lin helps Uncle Wu clean the house and do some cooking. Uncle Wu thanks Zhang Lin very much. He says, “Zhang Lin is a good boy. He is like my son.”