Creating More Convergent Series

and declare the series to converge provided that limN-,N SN exists as a finite real number, which number is called the sum of the given series. This paradigm becomes so ingrained in most of us that we seldom question its appropriateness. To be sure, a frequently useful theorem of K. Weierstrass (1815-1897) tells us that an absolutely convergent series may be rearranged at will, with the convergence undisturbed and the resulting sum the same. A companion theorem due to G. F. B. Riemann (1826-1866), asserting that a conditionally convergent series may be rearranged to obtain any sum in the extended reals, seems to lay to rest any desire to rearrange conditionally convergent series. But that desire has blossomed forth in the authors of this article. One of us learned, in the course of teaching an analysis class, that there exist permutations a of the positive integers N with the following two properties: