Cho dãy \(a_n=\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{n\left(n+1\right)}\). Hãy tính \(\lim\limits a_n\).
\(0\).\(\dfrac{1}{2}\).\(\dfrac{3}{2}\).\(1\).Hướng dẫn giải:Có \(\dfrac{1}{k\left(k+1\right)}=\dfrac{k+1-k}{k\left(k+1\right)}=\dfrac{1}{k}-\dfrac{1}{k+1}\) suy ra \(a_n=\dfrac{1}{1}-\dfrac{1}{n+1}\). Từ đó \(\lim\limits a_n=1-0=1.\)