Euler and Divergent Series

Abstract

Euler’s reputation is tarnished because of his views on divergent series. He believed that all series should have a value, not necessarily a limit as for convergent series, and that the value should remain invariant irrespective of the method of evaluation. Via the key concept of regularisation, which results in the removal of the infinity in the remainder of a divergent series, regularised values can be evaluated for elementary series outside their circles of absolute convergence such as the geometric series and for more complicated asymptotic series called terminants. Two different techniques for evaluating the regularised values are presented: the first being the standard technique of Borel summation and the second being the relatively novel, but more powerful, Mellin-Barnes regularisation. General forms for the regularised values of the two types of terminants, which vary as the truncation parameter is altered, are presented using both techniques over the entire complex plane. Then an extremely accurate and extensive numerical study is carried out for different values of the magnitude and argument of the main variable and the truncation parameter. In all cases it is found that the MB-regularised forms yield identical values to the Borel-summed forms, thereby vindicating Euler’s views and restoring his status as perhaps the greatest of all mathematicians.