\(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{1999\cdot2000}\)
\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{1999}-\dfrac{1}{2000}\)
\(=1-\dfrac{1}{2000}\)
\(=\dfrac{2000}{2000}-\dfrac{1}{2000}\)
\(=\dfrac{1999}{2000}\)
`1/1.2+1/2.3+1/3.4+⋯+1/1999.2000`
`=1/1-1/2+1/2-1/3+1/3-1/4+...+1/1999-1/2000`
`=1-1/2000`
`=1999/2000`
`1/1.2+1/2.3+1/3.4+⋯+1/1999.2000`
`=1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + .....+ 1/1999 - 1/2000`
` = 1 - 1/2000`
` = 2000/2000 - 1/2000`
` = 1999/2000`
`#BaoL i n h`
1/1.2 + 1/2.3 + 1/3.4 +..+ 1/1999.2000
=> 1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... - 1/1999 + 1/1999 - 1/2000
=> 1/1 - (- 1/2 + 1/2) + (- 1/3 + 1/3 ) + ...+ (- 1/1999 + 1/1999) - 1/2000
=> 1/1 + 0 + 0 +... + 0 - 1/2000
=> 1/1 - 1/2000
=> 2000/2000 - 1/2000
=> 1999/2000
\(=\)\(\dfrac{\text{1}}{\text{1}}\) \(-\dfrac{1}{2}\) \(+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+.....+\dfrac{1}{1999}-\dfrac{1}{2000}\)
\(=\dfrac{1}{1}-\dfrac{1}{2000}=\) \(\dfrac{\text{1999}}{2000}\)
