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]