On a Certain Sequence of Q U O T I E N T S of a Sequence

1. Example: Consider the Fibonacci sequence {xn} defined recursively by ar1 = 1, x2 = 1, xn + l = xn + xn_l if n > 2. It is well known that the q-sequence corresponding to {xn} converges to the real number ( 1 + /5)/2. This example shows that divergent sequences {xn} can have corresponding ^-sequences that converge. On the other hand, examples can be found of convergent sequences of quotients of convergent sequences. See Theorem 5 below. Whenever a sequence {xn} is defined recursively, say xY = a, x2 = b, xn+l = f(xn, xn_l) for n > 2 and positive numbers a and b, let S ( a, b , /) denote the corresponding qsequence, where f is a nonnegative function of two real variables which is defined and positive in the first quadrant and defined on the positive z/-axis. If iqn} converges, let z be its limit z = lim a .