Cho dãy số (un) với un= 1 /1.3 + 1/ 3.5 +...+ 1/ (2n−1).(2n+1) Khi đó lim un bằng?

\[\begin{array}{*{20}{l}}{{u_n} = \frac{1}{{1.3}} + \frac{1}{{3.5}} + ... + \frac{1}{{\left( {2n - 1} \right).\left( {2n + 1} \right)}}}\\{ = \frac{1}{2}.\left( {1 - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + ... + \frac{1}{{2n - 1}} - \frac{1}{{2n + 1}}} \right)}\\{ = \frac{1}{2}.\left( {1 - \frac{1}{{2n + 1}}} \right)}\\{ \Rightarrow \lim {u_n} = \lim \frac{1}{2}\left( {1 - \frac{1}{{2n + 1}}} \right) = \frac{1}{2}.}\end{array}\]