Eisenstein Series and Approximations to π

On page 211 in his lost notebook, in the pagination of [244], Ramanujan listed eight integers, 11, 19, 27, 43, 67, 163, 35, and 51 at the left margin. To the right of each integer, Ramanujan recorded a linear equation in Q ³ and R ². Although Ramanujan did not indicate the definitions of Q and R, we can easily (and correctly) ascertain that R and R are the Eisenstein series

$$Q(q) := 1+ 240\sum_{n=1}^{\i}\frac{n^3q^n}{1-q^n}$$

and

$$R(q) := 1- 504\sum_{n=1}^{\i}\frac{n^5q^n}{1-q^n}$$

, where $R(q) := 1- 504\sum_{n=1}^{\i}\frac{n^5q^n}{1-q^n},$. To the right of each equation in Q ³ and R ², Ramanujan entered an equality involving π and square roots. (For the integer 51, the linear equation and the equality involving π are not in fact, recorded by Ramanujan.)