We propose an introduction to the Laplace transform in which Riemann sums are used to approximate the expected net change in a function, assuming that it quantifies a process that can terminate at random. We assume only a basic understanding of probability.

Consider the series [image omitted] where the value of each a[subscript n] is determined by the flip of a coin: heads on the "n"th toss will mean that a[subscript n] =1 and tails that a[subscript n] = -1. Assuming that the coin is "fair," what is the probability that this "harmonic-like" series converges? After a moment's thought, many people…

This article describes how a presentation from the point of view of differential operators can be used to (partially) unify the myriad techniques in an introductory course in ordinary differential equations by providing students with a powerful, flexible paradigm that extends into (or from) linear algebra. (Contains 1 footnote.)