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 answer that the probability of convergence is 1. This is correct (though the proof is nontrivial), but it doesn't preclude the existence of a "divergent" example. Indeed, Feist and Naimi provided just such an example in 2004. In this paper, we construct an uncountably infinite family of examples as a companion result.