Divergent and Conditionally Convergent Series Whose Product is Absolutely Convergent

It has been shown by Abel that, if the product :of two conditionally convergent series :71-0 71 -0 is convergent, it converges to the product of their sums.Tests of the convergence of the product of conditionally convergent series have been worked out by A. Pringsheim,| A. Voss,J and myself.§ There exist certain conditionallyconvergent series which yield a convergent result when they are raised to a certain positive integral power, but which yield a divergent result when they are raised to a higher power.Thus,where r = 7/9, is a conditionally convergent series whose fourth power is convergent, but whose fifth power is divergent.|| These instances of conditionally