Ramanujan's series for 1/pi

Remarks

Convergence: by Stirling's formula $(4k)!/(k!)^4$ grows like a constant times $256^k k^{-3/2}$, while $396^4 \approx 2.46 \times 10^{10}$, so the ratio of successive terms tends to $256/396^4 \approx 1.04 \times 10^{-8}$. So each term adds about eight correct decimal digits. Example: the $k = 0$ term alone gives $\pi \approx 9801/(2\sqrt2 \cdot 1103) = 3.14159273\ldots$, against $\pi = 3.14159265\ldots$. Ramanujan published this series in 1914 among several series of the same shape, without a complete proof. Proof idea: the series comes from the theory of complete elliptic integrals and modular equations. It uses a singular value of the modulus attached to the imaginary quadratic field $\Q(\sqrt{-58})$, which is reflected in $e^{\pi\sqrt{58}} \approx 396^4 - 104$. Identities such as Legendre's relation convert the resulting relations between elliptic integrals into a formula for $1/\pi$. A complete proof was given later by Jonathan and Peter Borwein.

Going deeper

For a student

The series is the most famous member of the family of Ramanujan–Sato series, in which $1/\pi$ equals a sum of hypergeometric-type terms $\frac{(4k)!}{(k!)^4}(a + bk)x^k$ or similar. Each such series comes from a singular value of the modulus, where the elliptic curve has complex multiplication. The same idea with the $j$-invariant and the field $\Q(\sqrt{-163})$ gives the Chudnovsky algorithm, which gains about fourteen digits per term.