Question

Найдите сумму:1/1*2+1/2*3+1/3*4+.........+1/10000*10001

Answer 1

$\dfrac{1}{1\cdot 2}+\dfrac{1}{2\cdot 3}+\dfrac{1}{3\cdot 4}+...+\dfrac{1}{10000\cdot 10001}=\dfrac{2-1}{1\cdot 2}+\dfrac{3-2}{2\cdot 3}+\dfrac{4-3}{3\cdot 4}+\\ \\ \\ +....+\dfrac{10001-10000}{10000\cdot 10001}=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{10000}-\dfrac{1}{10001}=\\ \\ \\ =1-\dfrac{1}{10001}=\dfrac{10001-1}{10001}=\dfrac{10000}{10001}$