Question

cho\(A=\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{4.5}+.....+\frac{1}{99.100}\)

CMR:\(\frac{7}{12}< A< \frac{5}{6}\)

Answer 1

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

=( 1/1 + 1/2 +1/3 +1/4 + 1/5 + 1/6 +.....1/99 + 1/100) - 2(1/2 + 1/4 + 1/6 + .....+ 1/100)

=(1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 +.....+ 1/99 + 1/100) - ( 1 + 1/2 + 1/3 + .... + 1/50)

= 1/51 + 1/52 + 1/53 +....+ 1/100....>1/100

= ( 1/51 + 1/52 + 1/53 +.....+ 1/75) + ( 1/76 + 1/77 + 1/78 +.....+ 1/100)

Có 1/51>1/52>1/53>....>1/75 ; 1/76>1/77>1/78>....>1/100

A> 1/75.25 + 1/100.25= 1/3 + 1/4 = 7/12

A< 1/51.25+ 1/76.25 < 1/50.25 + 1/75.25= 1/2+1/3=5/6

Vậy 7/12< A< 5/6