B6a

bài 8a :
1/2^2 <1/1.2=1-1/2
……………………
1/9^2 <1/8.9=1/8-1/9
=>1/2^2+1/3^2+1/4^2+1/5^2+…+1/9^2 < 1-1/2+1/2-1/3+….+1/8-1/9=1-1/9=8/9
=>điều phải chứng minh

bài 8b :
Ta có :vế trái =1/1.2+1/3.4+1/5.6+….+1/99.100
=1-1/2+1/3-1/4+1/5-1/6+…+1/99-1/100
=(1+1/3+1/5+…+1/99)-(1/2+1/4+1/6+1/8+…+1/100)
=(1+13+1/5+…+1/99+1/2+1/4+1/6+1/8…+1/100)-(1/2+1/4+1/6+1/8+…+1/100).2
=(1+1/2+1/3+1/4+…+1/100)-(1+1/2+1/3+1/4+…+1/50)
=1/51+1/52+1/53+…+1/100=vế phải
Vậy đẳng thức được chứng minh

Bài 7:
A = 1 + 1/2^2 + 1/2^3 + ... + 1/2^2012
=> 2A = 2 + 1/2 + 1/2^2 + ... + 1/2^2011
=> 2A - A = (2 + 1/2 + 1/2^2 + ... + 1/2^2011) - (1 + 1/2^2 + 1/2^3 + ... + 1/2^2012)
<=> A = 1 + 1/2 - 1/(2^2012)
Vậy A = 3/2 - 1/(2^2012)

Bài 8:
a)
Ta có:
1/2^2 <1/1.2=1-1/2
……………………
1/9^2 <1/8.9=1/8-1/9
=>1/2^2+1/3^2+1/4^2+1/5^2+…+1/9^2 < 1-1/2+1/2-1/3+….+1/8-1/9=1-1/9 = 8/9 (đpcm)
b) 
VT = 1/1.2+1/3.4+1/5.6+….+1/99.100
=  1-1/2+1/3-1/4+1/5-1/6+…+1/99-1/100
=  (1+1/3+1/5+…+1/99)-(1/2+1/4+1/6+1/8+…+1/100)
=  (1+13+1/5+…+1/99+1/2+1/4+1/6+1/8…+1/100)-(1/2+1/4+1/6+1/8+…+1/100).2
=  (1+1/2+1/3+1/4+…+1/100)-(1+1/2+1/3+1/4+…+1/50)
= 1/51+1/52+1/53+…+1/100
Mà VP = 1/51+1/52+1/53+…+1/100 (đpcm)