Question

Aratati ca:
0,(6)<1/31+1/32+1/33+........1/90<2
1/2<1/1x2+1/2x3+1/3x4+....+1/2011x2012<1

Answer 1

a)
1/31+1/32+1/33+........1/90>1/90+1/90+1/90+........1/90=60/90=0,(6) 
=>   0,(6)<1/31+1/32+1/33+........1/90

1/31+1/32+1/33+........1/90<1/30+1/30+1/30+........1/30=60/30=2
1/31+1/32+1/33+........1/90<2


b)
  
1/1x2+1/2x3+1/3x4+....+1/2011x2012=1/1-1/2+1/2-1/3+1/3-1/4+....+1/2011-1/2012=1-1/2012=2011/2012

Cum  1/2<2011/2012<1   =>
1/2<1/1x2+1/2x3+1/3x4+....+1/2011x2012<1