(41C) and (42S) Arithmetic-Geometric Mean

[Albert Chan]

(07-29-2020 12:53 PM)Werner Wrote:  But the next GM in the sequence is not calculated as sqrt(lo*hi)? but as SQRT(GMi*AMi)

My mistake. I wrongly assumed AM converged to lo or hi too.

Proof (2nd attempt):

Again, assume GM sequence do not converge, but alternate between lo, hi
GM sequence = GM1, GM2, ..., lo, hi, lo, hi, ...

For big enough i, such that GM(i) = lo, and AM(i) converged to g:

GM(i+1) = hi = √(g*lo)
GM(i+2) = lo = √(g*hi)

If hi > lo, we have √(g*lo) > √(g*hi), which is impossible, even with rounding errors.

→ when AM converged, GM sequence is non-decreasing, will converge too.