scholarly journals A new method to sum divergent power series: educated match

2017 ◽  
Vol 1 (2) ◽  
pp. 025005 ◽  
Author(s):  
Gabriel Álvarez ◽  
Harris J Silverstone
Keyword(s):  
Power Series ◽  
New Method ◽  
Divergent Power Series

A new method for summation of divergent power series

1984 ◽  
Vol 25 (8) ◽  
pp. 2377-2389 ◽  
Author(s):  
Gustavo A. Arteca ◽  
Francisco M. Fernández ◽  
Eduardo A. Castro
Keyword(s):  
Power Series ◽  
New Method ◽  
Divergent Power Series

A note on the summation of divergent power series

1969 ◽  
Vol 66 (1) ◽  
pp. 109-114 ◽  
Author(s):  
R. E. Scraton
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Severe Restriction ◽  
Mathematical Problems ◽  
Divergent Power Series ◽  
Euler Transformation ◽  
Direct Use

Many mathematical problems which do not yield a closed-form solution admit of a solution in the form of a power series; differential equations are an obvious example. The direct use of this power series is limited to the interior of its circle of convergence, and this places a restriction—often a severe restriction—on its usefulness. The method described in this paper enables this restriction to be alleviated in many cases; it also enables the convergence of a power series within its circle of convergence to be improved. The method is based on the Euler transformation.

scholarly journals On the summability of divergent power series satisfying singular PDEs

2019 ◽  
Vol 357 (3) ◽  
pp. 258-262 ◽  
Author(s):  
Hua Chen ◽  
Zhuangchu Luo ◽  
Changgui Zhang
Keyword(s):  
Power Series ◽  
Divergent Power Series ◽  
Singular Pdes

Accurate power series for eigenvalues of spheroidal angle functions and their convergence radii

2011 ◽  
Vol 89 (11) ◽  
pp. 1083-1099
Author(s):  
Tam Do-Nhat
Keyword(s):  
Power Series ◽  
Least Squares ◽  
Complex Plane ◽  
Branch Point ◽  
Upper Bound ◽  
Least Squares Method ◽  
Radius Of Convergence ◽  
New Method ◽  
Recursion Relations

In this paper, the radius of convergence of the spheroidal power series associated with the eigenvalue is calculated without using the branch point approach. Studying the properties of the power series using the recursion relations among its coefficients in the new method offers some insights into the spheroidal power series and its associated eigenfunction. This study also used the least squares method to accurately compute the convergence radii to five or six significant digits. Within the circle of convergence in the complex plane of the parameter c = kF, where k is the wavenumber and F is the semifocal length of the spheroidal system, the extremely fast convergent spheroidal power series are computed with full precision. In addition, a formula for the magnitude of the upper bound of the error is obtained.

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