gọi biểu thức là A

ta có : 

A=3/1.2.3 + 5/2.3.4 +  7/3.4.5 +....+ 2017/1008.1009.1010

A= (1.2/1.2.3 + 2.2/2.3.4 + 3.2/3.4.5 + ... + 1008.2/1008.1009.1010) + (1/1.2.3 + 1/2.3.4 + 1/3.4.5 +...+ 1/1008.1009.1010)

A=(2/2.3 + 2/3.4 + 2/4.5 +...+ 2/1009.1010 + 1/2.(1/1.2-1/2.3+1/2.3-1/3.4+1/3.4-1/4.5 + ... + 1/1008.1009 - 1/1009.1010

A=2(1/2-1/3+1/3-1/4+1/4-1/5+...+1/1009-1/1010)+1/2.(1/2-1/1009.1/1010)

A<2.1/2 + 1/2.1/2 = 1+1/4 = 5/4 

OK nhớ tk cho mình nhé ( dấu này / là dấu phần nhé) chúc bạn học tốt

thank

Gọi biểu thức là \(A\). Ta có :

\(A=\dfrac{3}{1.2.3}+\dfrac{5}{2.3.4}+\dfrac{7}{3.4.5}+...+\dfrac{2017}{1008.1009.1010}\)

\(A=\left(\dfrac{1.2}{1.2.3}+\dfrac{2.2}{2.3.4}+\dfrac{3.2}{3.4.5}+...+\dfrac{1008.2}{1008.1009.1010}\right)+\left(\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+\dfrac{1}{3.4.5}+...+\dfrac{1}{1008.1009.1010}\right)\)\(A=\left(\dfrac{2}{2.3}+\dfrac{2}{3.4}+\dfrac{2}{4.5}+...+\dfrac{2}{1009.1010}\right)+\dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+\dfrac{1}{3.4}-\dfrac{1}{4.5}+...+\dfrac{1}{1008.1009}-\dfrac{1}{1009.1010}\right)\)

\(A=2\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{1009}-\dfrac{1}{1010}\right)+\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{1009.1010}\right)\)

\(A< 2.\dfrac{1}{2}+\dfrac{1}{2}.\dfrac{1}{2}=1+\dfrac{1}{4}=\dfrac{5}{4}\)