Calculate $$\lim_{n \to \infty} \sum_{k=1}^n \frac{\ln(1+\frac{1}{k})}{k(k+1)}.$$
I can use only the Squeeze Theorem or the Monotone Convergence Theorem or simple limit work; no big-o notation or integrals or whatever else. And no L'Hospital's rule!
I tried squeezing it between the first and the last term, but I got different results. Then I tried limiting it by the first term, but then I don't know how to solve the limit. Please help.