Given that:

F_m(x) = √(2 - √(2 + √(2 + √(2 + .....√(2 + √x))))) (m square roots)

Evaluate this limit:

Limit   F_m(2)/F_m(3)
m → ∞

It seems to approach 3/2. I think the later finding of sqrt(2) in the list below is erroneous based upon rounding errors, but maybe it's right:

 DEFDBL A-Z
CLS
x1 = 2
f1 = SQR(x1)
x2 = 3
f2 = SQR(x2)
FOR i = 1 TO 33
  f1 = SQR(2 + f1)
  f2 = SQR(2 + f2)
  PRINT SQR(2 - f1) / SQR(2 - f2)
NEXT

 

1.494645011073558
 1.498661325280844
 1.499665335813159
 1.499916334233543
 1.499979083575915
 1.499994770895146
 1.499998692721012
 1.499999673156927
 1.499999918364455
 1.499999979900743
 1.499999994338238
 1.5
 1.5
 1.49999998188236
 1.500000289882268
 1.499999710117676
 1.500004638125702
 1.500004638125702
 1.499925792168555
 1.499777365490425
 1.499703410773176
 1.502379065729704
 1.49071198499986
 1.58113883008419
 1.414213562373095

BTW it crashed at 25 due to division by "zero".

Edited on November 26, 2009, 1:49 pm

Posted by Charlie on 2009-11-26 13:47:50