newton[x_] := N[(x + 3/x)/2]

Nest[newton,  1.5, 5]

NestList[newton,  1.5, 5]

FixedPoint[newton,  1.5]

FixedPointList[newton,  1.5]

FixedPointList[newton,  1.5,  
      SameTest -> ((Abs[#1 - #2] < 10^-4)&)]