Cho un biết rằng un = 1/1.2

Ta luôn có: \(\frac{1}{{k\left( {k + 1} \right)}} = \frac{1}{k} - \frac{1}{{k + 1}}\) áp dụng vào \({u_n}\), ta có:

\({u_n} = \frac{1}{{1 \cdot 2}} + \frac{1}{{2 \cdot 3}} + ... + \frac{1}{{n\left( {n + 1} \right)}}\)

                        \( = \left( {\frac{1}{1} - \frac{1}{2}} \right) + \left( {\frac{1}{2} - \frac{1}{3}} \right) + \left( {\frac{1}{3} - \frac{1}{4}} \right) + ... + \left( {\frac{1}{n} - \frac{1}{{n + 1}}} \right) = 1 - \frac{1}{{n + 1}}\).

Do đó, \(\lim {u_n} = \lim \left( {1 - \frac{1}{{n + 1}}} \right) = 1\).