On Infinite Process Convergence, Part II: Brown's Series 1 + 1!x + 2!x2 + ?
|Title||On Infinite Process Convergence, Part II: Brown\'s Series 1 + 1!x + 2!x2 + ?|
|Author(s)||Thomas E Phipps|
|Keywords||Infinite Process Convergence, Brown's Series|
In our first essay on infinite process convergence [Phys. Essays 6, 135 (1993)] a method of ?terminal summation? of infinite series was introduced, which employed at each stage of the limiting process an asymptotic approximation to the remainder term at that stage. In application to two rapidly convergent (in the Cauchy sense) infinite series for , the method was shown to speed convergence. We now pass to the opposite extreme and apply the same method to a nowhere-convergent (in the Cauchy sense) series proposed by Brown [Phys. Essays 2, 270 (1989)]. As he conjectured, computable values of this function are found throughout the range of its real variable x. The evaluation proves to be highly computation-intensive. As previously claimed [Heretical Verities: Mathematical Themes in Physical Description (Classic Non-fiction Library, Urbana, IL, 1987)], the method accomplishes convergence-forcing as readily as it does convergence-speeding. A relationship of the function represented by the Brown series to the exponential integral along the real axis is shown ? on the basis of which a simplification of the definition of the latter, free of ?cuts,? is proposed. We offer Brown's series as a challenge to established ?summability? methods and flatly assert that none can match the convergence-forcing capabilities of terminal summation.