The Language of Finite Differences

Throughout the previous chapters, we derived results using infinitesimal calculus, such as the Euler-Maclaurin summation formula, the Boole summation formula, and the methods of summability of divergent series. In this chapter, we derive analogous results using the language of finite differences, as opposed to infinitesimal derivatives. Using finite differences and the theory of summability of divergent series, we present a simple pictorial proof to the Shannon-Nyquist sampling theorem. Finally, we derive many identities that relate to the Euler constant, the Riemann zeta function, the Gregory coefficients, and the Cauchy coefficients, among others.