Abstract: This work began with the rather simple task of estimating the difference between two consecutive "chunks" of the harmonic series. The result, achieved by calculus, suggested turning our focus to asymptotic expansions for chunks of zeta-like functions. We find such asymptotic expansions first using the classical Euler-Maclaurin summation and also with techniques specifically adapted to functions $f(N)$ that have asymptotic series in powers of $1/N.$ Since asymptotic expansions are unique, finding two apparently different such expansions leads to a sequence of new and old identities for Bernoulli polynomials. In the process of working through this phase, we were forced to come to grips with the variety of definitions and conventions for both the Bernoulli numbers and their related Bernoulli polynomials. One of our conclusions is that they all have their place and should probably all get reasonably equal treatment, as long as care is taken to come up with the right notation and to determine how they all relate to one another. Part of this presentation, then, is our contribution to the centuries old effort to "get this right." Naturally, we think we have done that, but so have many others before us! This is joint work with Gregory Galperin and Peter Andrews.