Вычислить сумму 1/1*10+1/10*20+1/20*30+.....................+1/1990*2000=

[latex] \frac{1}{1*10} = \frac{1}{10} \\ \frac{1}{10*20} = \frac{1}{10} (\frac{1}{10} - \frac{1}{20}) \\ \frac{1}{20*30} = \frac{1}{10} (\frac{1}{20} - \frac{1}{30}) \\ ... \\ \frac{1}{1990*2000} = \frac{1}{10}(\frac{1}{1990} - \frac{1}{2000})[/latex] [latex] \frac{1}{1*10} + \frac{1}{10*20} + \frac{1}{20*30} + ... + \frac{1}{1990*2000} = [/latex][latex]\frac{1}{10} + \frac{1}{10} (\frac{1}{10} - \frac{1}{20}) + \frac{1}{10} (\frac{1}{20} - \frac{1}{30}) + ... + \frac{1}{10} (\frac{1}{1990} - \frac{1}{2000} ) = [/latex][latex] \frac{1}{10} (1+ \frac{1}{10} - \frac{1}{20} + \frac{1}{20} - \frac{1}{30} + ... + \frac{1}{1990} - \frac{1}{2000} ) = [/latex][latex] \frac{1}{10} (1+ \frac{1}{10} - \frac{1}{2000} ) = \frac{1}{10} ( \frac{2000+200-1)}{2000} = \frac{2199}{20000} [/latex]