The Ramanujan and Chudnovsky series for $1/\pi$

Ramanujan's series (1914).

$\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^{\infty} \frac{(4k)!\,(1103 + 26390k)}{(k!)^4\, 396^{4k}},$

each term contributing roughly $8$ further correct decimal digits.

The Chudnovsky series (1988).

$\frac{1}{\pi} = 12 \sum_{k=0}^{\infty} \frac{(-1)^k (6k)!\,(13591409 + 545140134k)}{(3k)!\,(k!)^3\, 640320^{3k + 3/2}},$

each term contributing roughly $14$ further correct decimal digits. This is the series behind essentially every modern record computation of $\pi$.

Status: settled, but outside this library's stack. Both identities are proved theorems, not conjectures. They are not reachable here: they come from the theory of modular equations and modular forms, from singular values of the elliptic modulus, and, in the Chudnovsky case, from the class number one discriminant $-163$ that also produces the near-integer $e^{\pi \sqrt{163}}$. None of that machinery is developed in this library.

Not proved in this library. These identities are recorded with citations and are used nowhere.

What is known, and what would prove them here. Ramanujan stated seventeen series of this shape in his 1914 paper on modular equations and approximations to $\pi$, without proof; complete proofs were given much later, by J. M. and P. B. Borwein and independently by the Chudnovsky brothers, once the modular framework was in place. The general pattern is now understood as the Ramanujan-Sato family, indexed by levels and by the imaginary quadratic fields whose class numbers make the coefficients rational. What would discharge this item is a modular forms track: the modular group and its congruence subgroups, the $j$-invariant, complex multiplication and singular moduli. That is a long way outside a real analysis library.

Why it matters here. Together with the arithmetic-geometric mean algorithm, these series are the reason a reader should not conclude from the $\pi$ pages that this library has told the whole computational story. The formulas provable here converge slowly; the formulas actually used converge fast and rest on machinery from a different subject. Saying so explicitly is cheaper and more honest than silence.