S = $\frac{1}{2}$ + $\frac{5}{6}$ + $\frac{11}{12}$ + $\frac{19}{20}$ +$\frac{29}{30}$ + $\frac{41}{42}$ + $\frac{55}{56}$ + $\frac{71}{72}$ + $\frac{89}{90}$ = ( 1 - $\frac{1}{2}$) +( 1 - $\frac{5}{6}$) + ( 1 -$\frac{11}{12}$) + ( 1 - $\frac{19}{20}$) + ( 1 -$\frac{29}{30}$) + ( 1 - $\frac{41}{42}$) + ( 1 - $\frac{55}{56}$) +( 1 - $\frac{71}{72}$) + ( 1 -$\frac{89}{90}$) = (1+1+1+1+1+1+1+1+1) - ($\frac{1}{2}$ + $\frac{1}{6}$ + $\frac{1}{12}$ + $\frac{1}{20}$ +$\frac{1}{30}$ + $\frac{1}{42}$ + $\frac{1}{56}$ + $\frac{1}{72}$ + $\frac{1}{90}$) = 8 - ($\frac{1}{1.2}$ + $\frac{1}{2.3}$ + $\frac{1}{3.4}$ + $\frac{1}{4.5}$ +$\frac{1}{5.6}$ + $\frac{1}{6.7}$ + $\frac{1}{7.8}$ + $\frac{1}{8.9}$ + $\frac{1}{9.10}$)

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