Tìm x biết 1/1*3+1/3*5+1/5*7+…+1/(2x-1)(2x+1)

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1. 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:

   \(\begin{array}{l}
   \frac{1}{{1.3}} + \frac{1}{{3.5}} + \frac{1}{{5.7}} + …. + \frac{1}{{\left( {2x – 1} \right)\left( {2x + 1} \right)}}\\
    = \frac{1}{2}\left( {\frac{2}{{1.3}} + \frac{2}{{3.5}} + \frac{2}{{5.7}} + …. + \frac{1}{{\left( {2x – 1} \right)\left( {2x + 1} \right)}}} \right)\\
    = \frac{1}{2}\left( {1 – \frac{1}{3} + \frac{1}{3} – \frac{1}{5} + \frac{1}{5} – \frac{1}{7} + …. + \frac{1}{{2x – 1}} – \frac{1}{{2x + 1}}} \right)\\
    = \frac{1}{2}\left( {1 – \frac{1}{{2x + 1}}} \right)\\
    = \frac{1}{2}.\frac{{2x}}{{2x + 1}} = \frac{x}{{2x + 1}}
   \end{array}\)

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