A Distributional Approach to Conditionally Convergent Series

Whether the car’s gas tank is filled up on Monday and the paycheck is deposited on Tuesday, or vice versa, the contribution of those two transactions to the checkbook’s final balance is the same. By the commutative property, order does not matter for the algebraic addition of a finite number of terms. However, for a super banker who conducts an infinite number of transactions, order may matter. If a series (sum of all transactions/terms) is convergent and the order of term does not matter, then the series is absolutely convergent. If a series is convergent but the order of terms does matter, then it is conditionally convergent. Georg Bernhard Riemann proved the disturbing result that the final sum of a conditionally convergent series could be any number at all or divergent. In two, three and higher dimensions, the matter is even worse, and such series with double and triple sums are not even well-defined without first giving sum interpretation to the (standard) order in which the series is to be summed, e.g., in three dimensions, summing over expanding spheres or expanding cubes, whose points represent ordered triples occurring in the summation. In this note we show using elementary notions from distribution theory that an interpretation exists for conditionally convergent series so they have a precise, invariant meaning.