The Euler-Gompertz Constant and its relations to Wallis' Hypergeometric Series

Szilagyiova and A. (2016) The Euler-Gompertz Constant and its relations to Wallis' Hypergeometric Series. Master's Thesis / Essay, Mathematics.

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Abstract

Basic rules and definitions for summing divergent series, regularity, linearity and stability of a summation method. Examples of common summation methods: averaging methods, analytic continuation of a power series, Borel's summation methods. Introducing a formal totally divergent power series F(x)=0! - 1!x + 2!x^2 - 3!x^3 + ... The main interest is the value at x=1 called Wallis' hypergeometric series (WHS). Examine the four summation methods used by Euler to assign a finite value delta ~ 0.59 (Euler-Gompertz constant) to this series: (1) Solving an ordinary differential equation that has a formal power series solution F(x); (2) Repeated application of Euler transform - a regular summation method useful to accelerate oscillating divergent series; (4) Extrapolating a polynomial P(n) which formally gives WHS at n=0; (3) Expanding F(x) as a continued fraction and inspecting its convergence. Multiple connections among the four methods are established, mainly by notions of asymptotic series and Borel summability. The value of delta is approximated by 3 methods, at most to a precision of several thousand decimal places.

Item Type:	Thesis (Master's Thesis / Essay)
Degree programme:	Mathematics
Thesis type:	Master's Thesis / Essay
Language:	English
Date Deposited:	15 Feb 2018 08:26
Last Modified:	15 Feb 2018 08:26
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/14734

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