The c value you compute depends on computing the b value, though. It's not a recursion carried out in a way which avoids square roots. It's just carrying out th

The c value you compute depends on computing the b value, though. It's not a recursion carried out in a way which avoids square roots. It's just carrying out the same AM-GM sequence computation, and then taking a certain weighted harmonic mean over that sequence, which converges just because that original sequence converges anyway. Note that the arithmetic-harmonic mean I think you were going for is just the geometric mean (not the arithmetic-geometric mean, just the geometric mean simpliciter; see https://mathworld.wolfram.com/Arithmetic-HarmonicMean.html https://mathworld.wolfram.com/Arithmetic-HarmonicMean.html).