____1_______ + __1______ +____1__________ + .............+ _________ 1000*1019...

Решите задачу:

$\frac{1}{1000*1019}+\frac{1}{1019*1038}+\frac{1}{1038*1057}+...\frac{1}{1988*2007}=\\\\\frac{1019-1000}{1000*1019}*\frac{1}{19}+\frac{1038-1019}{1038*1019}*\frac{1}{19}+\\\\\+\frac{1057-1038}{1038*1057}*\frac{1}{19}+...+\frac{2007-1988}{1988*2007}*\frac{1}{19}=\\\\\frac{1}{19}*(\frac{1}{1000}-\frac{1}{1019}+\frac{1}{1019}-\frac{1}{1038}+\frac{1}{1038}-\frac{1}{1057}+...+\\\\\frac{1}{1998}-\frac{1}{2057})=\\\\\frac{1}{19}*(\frac{1}{1000}-\frac{1}{2007})=\frac{2007-1000}{1000*2007*19}=\frac{53}{2007000}$