A convergence sequence ==> it is a Cauchy sequence.

If not Cauchy then it does not converge.

Some CSs are more equal than others. Just see {1/n} and {(1 + x/n)^n}.

Is there a least convergent CS?

"least convergent CS" seems like a self-inflicted definition. Can you be more precise?

Hasn't anyone tried to rank-order CS by some measure of convergence? Here's two possible approaches:

1. How about defining a test that compares two CS to each other. For example, consider a function that is a derived sequence from a CS, epsilon(CS,i), that provides the sequence of epsilon values corresponding to the CS sequence values. This monotonic, non-increasing sequence will be CS, right? Next examine the ratio of two epsilon sequences and the inverse, epsilon(CS_1,i)/epsilon(CS_2,i) and epsilon(CS_2,i)/epsilon(CS_1,i). Is one a CS, is the inverse a CS, or might neither be CSes?

2. Or we could define various families of CS such as: a) CS that take linear time or index count to converge from epsilon to epsilon/2; b) CS that take quadratic time to converge from epsilon to epsilon/2; c) CS that take exponential time to converge from epsilon to epsilon/2, etc. Families might be rank ordered with respect to each other as with the computational complexity classes.