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 → ∞

(In reply to further computer exploration by Charlie)

I decided to explore this with a program on my graphing calculator as well.  I think it uses less precision so it crashes on division by zero at only 21.

Both f1 and f2 are approaching 2 as m (you used i) increases so eventually to computer cannot distinguish SQR(2-f2) from zero.  Your last term is likely an artifact.

Our programs start to deviate at about the 7th term due to the rounding differences.  My calculator only gives the 11th term as 1.5 not the 12th and 13th.

Posted by Jer on 2009-11-28 14:34:29