Conjecture a formula for $\sum_{k=1}^{n} \frac{1}{k(k+1)}=\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\cdots+\frac{1}{n(n+1)}$ from the value of this sum for small integers $n$. Prove that your conjecture is correct using mathematical induction.