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第75頁(yè)

信息發(fā)布者：

$=(1-\frac {1}{2})+(1-\frac {1}{6})+(1-\frac {1}{12})+...+(1-\frac {1}{110})$

$=1×10-(\frac {1}{2}+\frac {1}{6}+\frac {1}{12}+...+\frac {1}{110})$

$=10-(1-\frac {1}{2}+\frac {1}{2}-\frac {1}{3}+\frac {1}{3}-\frac {1}{4}+...+\frac {1}{10}-\frac {1}{11})$

$=10-(1-\frac {1}{11})$

$=9\frac {1}{11}$

$=\frac {5}{3}×\frac {3}{3×6}+\frac {5}{3}×\frac {3}{6×9}+\frac {5}{3}×\frac {3}{9×12}+...+\frac {5}{3}×\frac {3}{2997×3000}$

$=\frac {5}{3}×(\frac {1}{3}-\frac {1}{6}+\frac {1}{6}-\frac {1}{9}+\frac {1}{9}-\frac {1}{12}+...+\frac {1}{2997}-\frac {1}{3000})$

$=\frac {5}{3}×(\frac {1}{3}-\frac {1}{3000})$

$=\frac {5}{9}-\frac {1}{1800}$

$=\frac {111}{200}$

$=1-\frac {1}{3}+\frac {1}{3}-\frac {1}{5}+\frac {1}{5}-\frac {1}{7}+\frac {1}{7}-\frac {1}{9}$

$=1-\frac {1}{9}$

$=\frac {8}{9}$

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