Giải thích các bước giải:
Tổng quát:
\[\frac{1}{{n\left( {n + 2} \right)}} = \frac{1}{2}.\frac{2}{{n\left( {n + 2} \right)}} = \frac{1}{2}.\frac{{\left( {n + 2} \right) - n}}{{n\left( {n + 2} \right)}} = \frac{1}{2}.\left( {\frac{1}{n} - \frac{1}{{n + 2}}} \right)\]
Ta có:
\(\begin{array}{l}
\frac{1}{{1.3}} + \frac{1}{{3.5}} + \frac{1}{{5.7}} + .... + \frac{1}{{\left( {2x - 1} \right)\left( {2x + 1} \right)}}\\
= \frac{1}{2}\left( {\frac{2}{{1.3}} + \frac{2}{{3.5}} + \frac{2}{{5.7}} + .... + \frac{1}{{\left( {2x - 1} \right)\left( {2x + 1} \right)}}} \right)\\
= \frac{1}{2}\left( {1 - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + .... + \frac{1}{{2x - 1}} - \frac{1}{{2x + 1}}} \right)\\
= \frac{1}{2}\left( {1 - \frac{1}{{2x + 1}}} \right)\\
= \frac{1}{2}.\frac{{2x}}{{2x + 1}} = \frac{x}{{2x + 1}}
\end{array}\)
