Motivation for Ramanujan's mysterious $\pi$ formula

Question

The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$

Does anyone know how it works, or what the motivation for it is?

Answer 1

Here's an easy introduction to the basics, "Pi Formulas and the Monster Group".

http://sites.google.com/site/tpiezas/0013

Update: Just to make this more intriguing, define the fundamental unit $U_{29} = \frac{5+\sqrt{29}}{2}$ and fundamental solutions to Pell equations,

$$\big(U_{29}\big)^3=70+13\sqrt{29},\quad \text{thus}\;\;\color{blue}{70}^2-29\cdot\color{blue}{13}^2=-1$$

$$\big(U_{29}\big)^6=9801+1820\sqrt{29},\quad \text{thus}\;\;\color{blue}{9801}^2-29\cdot1820^2=1$$

$$2^6\left(\big(U_{29}\big)^6+\big(U_{29}\big)^{-6}\right)^2 =\color{blue}{396^4}$$

then we can see those integers all over the formula as,

$$\frac{1}{\pi} =\frac{2 \sqrt 2}{\color{blue}{9801}} \sum_{k=0}^\infty \frac{(4k)!}{k!^4} \frac{29\cdot\color{blue}{70\cdot13}\,k+1103}{\color{blue}{(396^4)}^k}$$

See also this MO post.

Answer 2

This is one of the most interesting results Ramanujan gave and it has a very deep and beautiful theory behind it. Most references regarding this formula try to treat it in high handed manner using modular forms. Ramanujan himself got this formula by remaining within the limits of real analysis and I have presented these ideas along with proofs in my blog post.

Please note that the actual calculation to obtain the numbers 1103 and 26390 in the formula is difficult. Especially no one knows how Ramanujan got 1103 and modern approach to get 1103 is based on numerical calculations.

By Ramanujan's theory (explained in my blog post linked above) we can find infinitely many series of the form $$\frac{1}{\pi} = \sum_{n = 0}^{\infty}(a + bn)d_{n}c^{n}\tag{1}$$ where $a, b, c$ are certain specific algebraic numbers and $d_{n}$ is some sequence of rationals usually expressed in terms of factorials. The modern theory of modular forms allows us to get more details about their algebraic nature (say for example we can get the degree of minimal polynomials of $a, b, c)$. In the case of the current formula it can be shown that both $a, b$ must be quadratic irrationals and $c$ turns out to be a rational number. The calculation of $b, c$ is possible by formulas given by Ramanujan. It is the value of $a$ (related to $1103$) which is difficult to obtain. Now the modern approach goes like this. Since we know the value of $b, c$ and $\pi$ (via some other series calculation) we can find the numerical value of $a$. Knowing that it is a quadratic irrational we can search for integers $p, q, r$ such that $a$ is a root of $px^{2} + qx + r = 0$. This way the quadratic equation is found and the root $a$ is then evaluated in algebraic form.

There are direct formulas to calculate $a, b, c$ and we have two forms of such formulas. One of the forms is a finite formula which may require computations of algebraic nature (so that effectively the value is expressible as a radical expression). Another formula is kind of based on infinite series/product approach which can give numerical values of $a, b, c$. While the algebraic formula for $b, c$ is easy to calculate, the algebraic formula for $a$ is very difficult to compute. Hence the modern approach relies on numerical calculation of $a$. But I very strongly suspect that Ramanujan being an expert in radical manipulation must have found the algebraic value of $a$ using a direct radical manipulation.

In this regard also try to read the book "Pi and the AGM" by Borwein Brothers as they are the first ones to prove this formula of Ramanujan. Also see this answer on mathoverflow for calculation of the constant $1103$.

@Derek Jennings

The general series given in MathWorld is the one discovered by Chudnovsky brothers and it is a different series based on Ramanujan's ideas, but the series in the question under discussion can not be obtained from this general formula of Chudnovsky. A proof of this general series of Chudnovsky is presented in my blog post.

Answer 3

Here is a nice article Entitled: "Ramanujan's Series for $\displaystyle\frac{1}{\pi}$ : A Survey", by Bruce C.Berndt. This article appeared in the American Mathematical Monthly *August/September* 2009. You can see it here.

• http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.158.2533&rep=rep1&type=pdf

Answer 4

The explanation for the existence of this series is given here. Search for the phrase "The general form of the series is" to locate it. The series cited in the question appears immediately before the explanation.