Reconstructing and Extending Ramanujan's 1914 Series for 1/π: A Line-by-Line Proof of the Modular Equation Theory, With Five New Quartic-Theory Series

This paper has two parts. First, we give a complete, self-contained reconstruction of the proof of Equation 3 in Ramanujan's 1914 paper "Modular equations and approximations to π" [Ramanujan 1914], starting from the elementary problem of the perimeter of an ellipse and building up the theory of Jacobi theta functions, the nome, and the Jacobi triple product needed to prove it. We then reconstruct the general method of Section 13 of the same paper — the production of series for 1/π from singular moduli — and verify it exactly against Ramanujan's own published examples for n = 3 and n = 7. Second, we present five new series for 1/π, in the quartic ("K1") theory of Ramanujan's Section 14, associated with an unfinished two-column table in Ramanujan's notebooks whose entries for n = 7, 15, 23, 35, 71 were left unexplained (Berndt notes: "it is unclear to us why Ramanujan studied this particular function" [Berndt 1998, Sec. 34.10]). Each of the five series is verified numerically to more than 40 decimal digits of agreement with π. Two of the five (n = 7 and n = 35) are shown to match series previously published by other authors in different closed forms; the remaining three (n = 15, 23, 71) are not found in the literature searched for this paper. The analytic derivation connecting the singular moduli to these five closed forms — the quartic-theory analogue of Ramanujan's Equations 25–27 — is not carried out here and is identified explicitly as open work.