Ta có:
7/12 = 4/12 + 3/12 = 1/3 + 1/4 = 20/60 + 20/80
1/41 + 1/42 + 1/43 +...+ 1/79 + 1/80 = (1/41 + 1/42 + 1/43 + ...+ 1/60) + (1/61 + 1/62 +...+ 1/79 + 1/80)
Do 1/41> 1/42 > 1/43 > ...>1/59 > 1/60
=> (1/41 + 1/42 + 1/43 + ...+ 1/60) > 1/60 + ...+ 1/60 = 20/60
và 1/61> 1/62> ... >1/79> 1/80
=> (1/61 + 1/62 +...+ 1/79 + 1/80) > 1/80 + ...+ 1/80 = 20/80
Vậy: 1/41 + 1/42 + 1/43 +...+ 1/79 + 1/80 > 20/60 + 20/80 = 7/12
=> 1/41 + 1/42 + 1/43 +...+ 1/79 + 1/80 > 7/12
*.Ta có lại có:
5/6 = 2/6 + 3/6 = 1/3 + 1/2 = 20/60 + 20/40
1/41 + 1/42 + 1/43 +...+ 1/79 + 1/80 = (1/41 + 1/42 + 1/43 + ...+ 1/60) + (1/61 + 1/62 +...+ 1/79 + 1/80)
Do 1/40> 1/41> 1/42 > 1/43 > ...>1/59 > 1/60
=> 1/40+1/40+…+1/40= 20/40>(1/41 + 1/42 + 1/43 + ...+ 1/60)
và 1/60> 1/61> 1/62> ... >1/79
=> 1/60+1/60+….+1/60= 20/60>(1/61 + 1/62 +...+ 1/79 + 1/80)
Vậy: 20/40+20/60=5/6> 1/41 + 1/42 + 1/43 +...+ 1/79 + 1/80
=> 1/41 + 1/42 + 1/43 +...+ 1/79 + 1/80 < 5/6
Vậy: 7/12 < 1/41 + 1/42 + 1/43 + ... + 1/79 + 1/80 < 5/6