dấu này là mũ hay là gì ? ^^^^^

A = 1/1.3 - 1/2.4 + 1/3.5 - 1/4.6 + ....+ 1/97.99 - 1/98.100

A = (1/1.3 + 1/3.5 + ...+ 1/97.99) - (1/2.4 - 1/4.6 + ...+ 1/98.100)

Đặt B = 1/1.3 + 1/3.5 + ..+ 1/97.99

C = 1/2.4 + 1/4.6 + ...+ 1/98.100

2B = 2/1.3 + 2/3.5 + ...+ 2/97.99

2B = 1/1 - 1/3 + 1/3- 1/5 + .. + 1/97 - 1/99

2B = 1/1 - 1/99

2B = 98/99

B = 98/99 : 2

B = 49/99

C = 1/2.4 + 1/4.6 + ... + 1/98.100

2C = 2/2.4 + 2/4.6 + ... + 2/98.100

2C = 1/2 - 1/4 + 1/4 - 1/6 + ... + 1/98 - 1/100

2C = 1/2 - 1/100

2C = 49/100

C = 49/100 : 2

C = 49/200

A = B - C

A = 49/99 - 49/200

B = 9800/19800 - 4851/19800

B = 4949/19800

=1-1/3-1/2+1/4+1/3-1/5-1/4+1/6+...+1/97-1/99-1/98+1/100

=1-1/2-1/99-1/98=2327/4851

\(C=\dfrac{4}{1.3}.\dfrac{9}{2.4}.\dfrac{16}{3.5}.\dfrac{25}{4.6}....\dfrac{9801}{9800}=\)

\(=\dfrac{2^2.3^2.4^2.5^2.....99^2}{1.2.3^2.4^2.5^2....98^2.99.100}=\dfrac{2.99}{100}=\dfrac{198}{100}=1,98\)

\(\frac{1}{1.3}-\frac{1}{2.4}+\frac{1}{3.5}-\frac{1}{4.6}+...+\frac{1}{97.99}-\frac{1}{98.100}\)

\(=1-\frac{1}{3}-\frac{1}{2}+\frac{1}{4}+\frac{1}{3}-\frac{1}{5}-\frac{1}{4}+\frac{1}{6}+...+\frac{1}{97}-\frac{1}{99}-\frac{1}{98}+\frac{1}{100}\)

\(=1-\frac{1}{2}-\frac{1}{99}-\frac{1}{98}\)

\(=\frac{2327}{4851}\)

Đặt A=1/1.3 - 1/2.4 +1/3.5 -1/4.6 +.....+1/97.99 -1/98.100

     4A= 4/1.3 -4/2.4 +4/3.5 -4/4.6 +.....+4/97.99 -4/98.100

          =(4/1.3 +4/3.5 +...+4/97.99) - (4/2.4 +4/4.6 +...+4/98.100)

          =(1/1 -1/3+1/3-1/5+...+1/97-1/99)-(1/2 -1/4 -....1/98-1/100)

         =(1/1-1/99)-(1/2-1/100)

         4A=98/99 - 99/100

         A= (98/99-99/100) :4

Đặt A=\(\dfrac{1}{1.3}+\dfrac{1}{2.4}+\dfrac{1}{3.5}+...+\dfrac{1}{98.100}\)

A=\(\left(\dfrac{1}{1.3}+...+\dfrac{1}{97.99}\right)+\left(\dfrac{1}{2.4}+...+\dfrac{1}{98.100}\right)\)

A=\(\left(\dfrac{1}{1}-\dfrac{1}{99}\right)+\left(\dfrac{1}{2}-\dfrac{1}{100}\right)\)

A=\(\dfrac{98}{99}-\dfrac{49}{100}\)

A=\(\dfrac{4949}{9900}\)

Mà \(\dfrac{3}{4}=\dfrac{7425}{9900}\)

Vậy A<\(\dfrac{3}{4}\)

Bạn hãy tính \(\dfrac{1}{1.3}+...+\dfrac{1}{98.100}\)= \(\dfrac{4949}{9900}\) sau đo chỉ cần chứng minh nó nhỏ hơn bằng cách quy đồng .