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We derive analytic solutions for the potential and field in a one-dimensional system of masses or charges with periodic boundary conditions, in other words, Ewald sums for one dimension. We also provide a set of tools for exploring the system evolution and show that it is possible to construct an efficient algorithm for carrying out simulations. In the cosmological setting we show that two approaches for satisfying periodic boundary conditions-one overly specified and the other completely general-provide a nearly identical clustering evolution until the number of clusters becomes small, at which time the influence of any size-dependent boundary cannot be ignored. Finally, we compare the results with other recent work with the hope to provide clarification over differences these issues have induced.