  We present an easily defined countable family of permutations of the natural
numbers for which explicit rearrangements (i.e., the sums induced by the
permutations) can be computed. The digamma function proves to be the key tool
for the computations found here for the alternating harmonic series. The
permutations $\phi$ under consideration are simple in a sense: they are
involutions ($\phi\circ\phi$ is the identity function). We show that the
countable set of rearrangements obtained from the simple involutions considered
below are dense in the reals.
