Every real number $x,0\<x\le 1$, has an essentially unique expansion as a Pierce series :$$x=\frac{1}{x^1}-\frac{1}{x^1{x}^2}+\frac{1}{x^1{x}^2{x}^3}-\cdots$$where the $x_i$ form a strictly increasing sequence of positive integers. The expansion terminates if and only if $x$ is rational. Similarly, every positive real number $y$ has a unique expansion as an Engel series :$$y=\frac{1}{y^1}-\frac{1}{y^1{y}^2}+\frac{1}{y^1{y}^2{y}^3}+\cdots$$where the $y_i$ form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi be not eventually constant. Again, such an expansion terminates...