Sums of Powers, Bernoulli Numbers, and the Gamma Function

The principal topics in Chapter 7 concern sums of powers, an extended definition of Bernoulli numbers, the Riemann zeta-function ζ(s) and allied functions, Ramanujan’s theory of divergent series, and the gamma function. This chapter thus represents a continuation of the subject matter of Chapters 5 and 6. Perhaps more so than any other chapter in the second notebook, Chapter 7 offers a considerable amount of numerical calculation. The extent of Ramanujan’s calculations is amazing, since he evidently performed them without the aid of a mechanical or electrical device.