On rewriting we have the series as,
$\frac{1}{{1 \times 2 \times 3}} + \frac{1}{{2 \times 3 \times 4}} + …………….. + \frac{1}{{100 \times 101 \times 102}}$
$ = \sum\limits_{r = 1}^{100} {\frac{1}{{r(r + 1)(r + 2)}}} $
$ = \frac{1}{2}\sum\limits_{r = 1}^{100} {\left[ {\frac{1}{{r(r + 1)}} – \frac{1}{{(r + 1)(r + 2)}}} \right]} $ Continue reading $\scriptscriptstyle \frac{1}{{102 \times 101 \times 100}} + \frac{1}{{101 \times 100 \times 99}} + ……… + \frac{1}{{3 \times 2 \times 1}} = ?$
${t_r} = \frac{1}{{4{r^2} – 1}}$
$ = \frac{1}{{(2r – 1)(2r + 1)}} = \frac{1}{2}\left( {\frac{1}{{2r – 1}} – \frac{1}{{2r + 1}}} \right)$
${S_n} = \sum\limits_{r = 1}^n {\frac{1}{2}\left( {\frac{1}{{2r – 1}} – \frac{1}{{2r + 1}}} \right)} $
${S_n} = \frac{1}{2}\left[ {\left( {\frac{1}{1} – \frac{1}{3}} \right) + \left( {\frac{1}{3} – \frac{1}{5}} \right) + \left( {\frac{1}{5} – \frac{1}{7}} \right) + ……. + \left( {\frac{1}{{2n – 1}} – \frac{1}{{2n + 1}}} \right)} \right]$
${S_n} = \frac{1}{2}\left[ {1 – \frac{1}{{2n + 1}}} \right] = \frac{n}{{2n + 1}}$
