Cho dãy số \[({u_n})\]với \[{u_n} = \frac{1} + \frac{1} + ... + \frac{1}{{\left( {2n - 1} \right).\left( {2n + 1} \right)}}\] Khi đó \[lim\,{u_n}\] bằng?

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

Đáp án cần chọn là: A