Thực hiện phép tính (1 - 1 / (1 + 2)) (1 - 1/(1 + 2 + 3))

Ta xét tổng quát:

\(1 - \frac{1}{{1 + 2 + 3 + ... + n}} = 1 - \frac{1}{{\frac{{n.\left( {n + 1} \right)}}{2}}} = 1 - \frac{2}{{n\left( {n + 1} \right)}} = \frac{{{n^2} + n - 2}}{{n\left( {n + 1} \right)}} = \frac{{\left( {n - 1} \right)\left( {n + 2} \right)}}{{n\left( {n + 1} \right)}}\)

Khi đó ta có:

\(\left( {1 - \frac{1}{{1 + 2}}} \right)\left( {1 - \frac{1}{{1 + 2 + 3}}} \right)...\left( {1 - \frac{1}{{1 + 2 + 3 + ... + 2006}}} \right)\)

\( = \frac{{1.4}}{{2.3}}.\frac{{2.5}}{{3.4}}....\frac{{2005.2008}}{{2006.2007}}\)

\( = \frac{{\left( {1.2.3...2005} \right).\left( {4.5.6....2008} \right)}}{{\left( {2.3.4....2006} \right).\left( {3.4.5....2007} \right)}} = \frac{{1.2008}}{{2006.3}} = \frac{{2008}}{{6018}}\).