ramanujan's Π summation

When G.H. Hardy first began his correspondence with Srinivasa Ramanujan, he was amazed by the prodigy's elegant and surprising series for irrational and transcendental values. Although Ramanujan hadn't included proofs of the theorems, Hardy told his colleagues, "they must be true, because, if they were not true, no one would have the imagination to invent them."

For example, here's Ramanujan's sum for Π which, after just 3 terms, overwhelms IEEE 754 double precision. Modern discoveries, like spigot algorithms on non-decimal bases, can find a given individual digit in constant time. But for fully expanding Π, no one has surpassed Ramanujan's expression: both the Chudnovsky brothers and Yasumasa Kanada adapted it to achieve their milestone calculations.