Accelerated series for universal constants, by the WZ method

In [1] Amdeberhan and Zeilberger have given a general method, based on WZ theory, for finding rapidly converging series for universal constants. We give another, somewhat different method here. In the form that we shall give to the method, the summand will satisfy a first order homogeneous recurrence but the sum will satisfy a first order inhomogeneous recurrence. What we obtain are remarkable families of representations of the constants, one for each n 0. If we look at eq. (5) below, for instance, we see that the constant 2=6 is equal to any one of infinitely many series, one for each n, one of which converges geometrically rapidly.