  Extending work of Foster, Doyle, and others, we show how the Foster Theorems,
a family of results concerning effective resistances on finite graphs, can in
certain cases be extended to infinite graphs. A family of sum rules is then
obtained, which allows one to easily calculate the sum of the resistances over
all paths of a given length. The results are illustrated with some of the most
common grids in the plane, including the square, triangular, and hexagonal
grids.
