Найдите последовательно значение каждой из разностей:

$\frac{1}{2} - \frac{1}{3} = \frac{3-2}{6} = \frac{1}{6}$
$\frac{1}{3} - \frac{1}{4} = \frac{4-3}{12} = \frac{1}{12}$
$\frac{1}{4} - \frac{1}{5} = \frac{5-4}{20} = \frac{1}{20}$
$\frac{1}{5} - \frac{1}{6} = \frac{6-5}{30} = \frac{1}{30}$
$\frac{1}{6} - \frac{1}{7} = \frac{7-6}{42} = \frac{1}{42}$
$\frac{1}{7} - \frac{1}{8} = \frac{8-7}{56} = \frac{1}{56}$
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$\frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{42} + \frac{1}{56} = \frac{2+1}{12} + \frac{1}{20} + \frac{1}{42} + \frac{1}{56} = \frac{1}{4} + \frac{1}{20} + \frac{1}{42} + \frac{1}{56} = \frac{5+1}{20} + \frac{3}{10} + \frac{1}{42} + \frac{1}{56} = \frac{21+5}{210} + \frac{1}{56} = \frac{26}{210} + \frac{1}{56} = \frac{52+7.5}{420} = \frac{59.5}{420}$