Ramanujan's "Most Beautiful Identity"

You may have heard of Ramanujan (it is hard to believe you haven’t), but you may know little or nothing of his work. Of all the many identities he presented, Hardy chose one which for him represented the best of Ramanujan. I would like to show you this identity, and prove it. Following Euler, we define a partition of the positive integer n as a representation of n as a sum of positive integers, in which order is unimportant. The partitions of 4 are 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1. The number of partitions of n is denoted by p(n); thus, p(4) = 5. For convenience, we define p(0) = 1. Euler showed that the partition generating function