Summation of divergent series from the nonstandard point of view
The authors perform a development of techniques of computations using nonstandard analysis. The known formula \(1+\tau +\tau ^2+\dots =\frac {1}{1-\tau}\) valid for \(\tau\) s.t. \(|\tau |<1\) and similar formulas are adapted also for \(\tau\) being the shift operator (raising the indices of the series by 1; to the series \(a_0+a_1+a_2+\dots\) the series \(b_0+b_1+b_2+\dots \) is assigned, where \(b_i=a_{i+1}\)). In this framework they describe the Hutton and Euler transform of series used for the summation of some divergent series. They show the necessity of adaption of the understanding of \(1\) (as a suitable matrix near to the unit matrix) and exponentiation (also in a nonstandard manner). It is obvious that they use matrix calculus for matrices of size \(\eta \times \eta\) (\(\eta\) being infinitely large) and standard part (shadow) technique. The paper is written in a very nice manner. The authors bring the reader's attention to some problems connected with the nonstandard work, too. I think that the paper gives a lot of motivations for thinking and I heartily recommend it for reading.
scientific article; zbMATH DE number 1095842