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   $\dfrac{1}{2}$ $\ +$ $\dfrac{5}{6}$ $\ +$ $\dfrac{11}{12}$ $\ +$ $\dfrac{19}{20}$ $\ +$ $\dfrac{41}{42}$ $\ +$ $\dfrac{55}{56}$ $\ +$ $\dfrac{71}{72}$ $\ +$ $\dfrac{89}{90}$

= ( 1 - $\dfrac{1}{2}$ ) + ( 1 - $\dfrac{1}{6}$ ) + ( 1 - $\dfrac{1}{12}$ ) + ( 1 - $\dfrac{1}{20}$ ) + ( 1 - $\dfrac{1}{42}$ ) + ( 1 - $\dfrac{1}{56}$ ) + ( 1 - $\dfrac{1}{72}$ ) + ( 1 - $\dfrac{1}{90}$ )

= $\ ( 1 + 1 + 1 + 1 + 1 + 1+ 1 + 1 + 1 )$ - ( $\dfrac{1}{2}$ + $\dfrac{1}{6}$ + $\dfrac{1}{12}$ + $\dfrac{1}{20}$ + $\dfrac{1}{42}$ + $\dfrac{1}{56}$ + $\dfrac{1}{72}$ + $\dfrac{1}{90}$ )

= $\ ( 1 × 9 )$ - ( $\dfrac{1}{1 × 2}$ + $\dfrac{1}{2 × 3}$ + $\dfrac{1}{3 × 4}$ + $\dfrac{1}{4 × 5}$ + $\dfrac{1}{5 × 6}$ + $\dfrac{1}{6 × 7}$ + $\dfrac{1}{7 × 8}$ + $\dfrac{1}{8 × 9}$ + $\dfrac{1}{9 × 10}$ )

= $\ 9$ - ( $\dfrac{1}{1}$ - $\dfrac{1}{2}$ + $\dfrac{1}{2}$ - $\dfrac{1}{3}$ + $\dfrac{1}{3}$ - $\dfrac{1}{4}$ + $\dfrac{1}{14}$ - $\dfrac{1}{5}$ + $\dfrac{1}{5}$ - $\dfrac{1}{6}$ + $\dfrac{1}{6}$ - $\dfrac{1}{7}$ + $\dfrac{1}{7}$ - $\dfrac{1}{8}$ + $\dfrac{1}{8}$ - $\dfrac{1}{9}$ + $\dfrac{1}{9}$ - $\dfrac{1}{10}$ )

= $\ 9$ - ( $\dfrac{1}{1}$ - $\dfrac{1}{10}$ )

= $\ 9$ - $\dfrac{9}{10}$

= $\dfrac{81}{10}$ = $\ 8,1$