Ramanujan's Series for 1/π: A Survey

When we pause to reflect on Ramanujan’s life, we see that there were certain events that seemingly were necessary in order that Ramanujan and his mathematics be brought to posterity. One of these was V. Ramaswamy Aiyer’s founding of the Indian Mathematical Society on 4 April 1907, for had he not launched the Indian Mathematical Society, then the next necessary episode, namely, Ramanujan’s meeting with Ramaswamy Aiyer at his office in Tirtukkoilur in 1910, would also have not taken place. Ramanujan had carried with him one of his notebooks, and Ramaswamy Aiyer not only recognized the creative spirit that produced its contents, but he also had the wisdom to contact others, such as R. Ramachandra Rao, in order to bring Ramanujan’s mathematics to others for appreciation and support. The large mathematical community that has thrived on Ramanujan’s discoveries for nearly a century owes a huge debt to V. Ramaswamy Aiyer. 1. THE BEGINNING. Toward the end of the first paper [57], [58 ,p . 36] that Ramanujan published in England, at the beginning of Section 13, he writes, “I shall conclude this paper by giving a few series for 1/π.” (In fact, Ramanujan concluded his paper a couple of pages later with another topic: formulas and approximations for the perimeter of an ellipse.) After sketching his ideas, which we examine in detail in Sections 3 and 9, Ramanujan records three series representations for 1/π .A s is customary, set