This talk is dedicated to the memory of Srinivasa Ramanujan, the self-taught Indian genius of mathematics, who died a century ago. Among his achievements is a way to assign finite values to divergent series, but he hardly left any explanation about it. Following the few hints that he gave, we reconstruct a justification for his approach, which provides correct values of the Riemann zeta function with negative arguments. We further discuss physical application of Ramanujan summation in the context of the Casimir effect, where his way of removing a divergent constant corresponds to the renormalization of the photon energy density. This leads to the prediction of the Casimir force between conducting plates, which has now been accurately confirmed by experiments.