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By iterations here we mean either continued-fraction recurrence steps, or series-summand additions. 48 Algorithm for Rη (a, b) with real η, a, b > 0: 1. 1); 2. 3) or some other scheme such as rapid ψ computations; } 3. 1); } 4. 2), √ 2R1 (a + b)/2, ab − R1(b, a). c ∗ Say, |1 − b/a| > ε > 0 for any fixed ε > 0. 49 • It is an implicit tribute to Ramanujan’s ingenuity that the final step (4) of the algorithm allows entire procedure to go through for all positive real parameters. 2), but Ramanujan’s AGM identity is finer!

2) yields 1 R(1) = log 2 = , 1 1+ 2+ 1 1 3+ 1 + ... but alas the beginnings of this fraction are misleading; subsequent elements an run 2 1 2 log 2 = [1, 2, 3, 1, 5, , 7, , 9, , . . ], 3 2 5 being as αn = n, 4/n resp. for n odd, even. Similarly, one can derive 2 − log 4 = [13, r2, 23, r4, 33, r6, 43, . . ], where the even-indexed fraction elements r2n are computable rationals. 33 • Though these RCFs do not have integer elements, the growths of the αn provide a clue to the convergence rate, which we study in a subsequent section.

The exponent h(a) can be taken to be c0 min(1, 4π 2/a2) where the constant c0 is absolute. Remark: While the bound is computationally poor, as noted, convergence does occur. • With effort, the exponent h(a) can be sharpened— and made more explicit. • Indeed, for a = b or even a ≈ b we now have many other, rapidly convergent options. 45 Proof: With a view to induction, assume that for some constants (n-independent) d(a), g(a) and for n ∈ [1, N − 1] we have qn < dng . 1) mean that An > f (a)/n for an n-independent f .

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