Đáp án:
\[\frac{{ - 2011}}{{2013}}\]
Giải thích các bước giải:
Ta có:
\[\frac{1}{{n\left( {n + 1} \right)}} = \frac{{\left( {n + 1} \right) - n}}{{n\left( {n + 1} \right)}} = \frac{{n + 1}}{{n\left( {n + 1} \right)}} - \frac{n}{{n\left( {n + 1} \right)}} = \frac{1}{n} - \frac{1}{{n + 1}}\]
Áp dụng đẳng thức trên ta có:
\[\begin{array}{l}
\frac{1}{{2013}} - \frac{1}{{2013.2012}} - \frac{1}{{2012.2011}} - .... - \frac{1}{{3.2}} - \frac{1}{{2.1}}\\
= \frac{1}{{2013}} - \left( {\frac{1}{{2012}} - \frac{1}{{2013}}} \right) - \left( {\frac{1}{{2011}} - \frac{1}{{2012}}} \right) - .... - \left( {\frac{1}{2} - \frac{1}{3}} \right) - \left( {1 - \frac{1}{2}} \right)\\
= \frac{1}{{2013}} - \frac{1}{{2012}} + \frac{1}{{2013}} - \frac{1}{{2011}} + \frac{1}{{2012}} - \frac{1}{{2010}} + \frac{1}{{2011}} - ..... - \frac{1}{2} + \frac{1}{3} - 1 + \frac{1}{2}\\
= \frac{2}{{2013}} - 1 = \frac{{ - 2011}}{{2013}}
\end{array}\]
