scholarly journals A Natural Series for the Natural Logarithm

10.37236/880 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Oliver T. Dasbach
Keyword(s):  
Power Series ◽  
Infinite Series ◽  
Mahler Measure ◽  
Series Expansions ◽  
Natural Logarithm ◽  
Similar Series ◽  
Power Series Expansions ◽  
Gromov Norm ◽  
Logarithm Function ◽  
Natural Series

Rodriguez Villegas expressed the Mahler measure of a polynomial in terms of an infinite series. Lück's combinatorial $L^2$-torsion leads to similar series expressions for the Gromov norm of a knot complement. In this note we show that those formulae yield interesting power series expansions for the logarithm function. This generalizes an infinite series of Lehmer for the natural logarithm of $4$.

scholarly journals Some interesting series arising from the power series expansion of(sin-1x)q

2005 ◽  
Vol 2005 (14) ◽  
pp. 2329-2336 ◽  
Author(s):  
Habib Muzaffar
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Infinite Series ◽  
Power Series Expansion ◽  
Series Expansions ◽  
Power Series Expansions

Starting from the power series expansions of(sin-1x)q, for1≤p≤4, formulae are obtained for the sum of several infinite series. Some of these evaluations involveζ(3).

scholarly journals Parallel Software to Offset the Cost of Higher Precision

2021 ◽  
Vol 40 (2) ◽  
pp. 59-64
Author(s):  
Jan Verschelde
Keyword(s):  
Power Series ◽  
Parallel Algorithms ◽  
Series Expansions ◽  
Use Case ◽  
Double Precision ◽  
Algebraic Space ◽  
Space Curves ◽  
Computational Overhead ◽  
Power Series Expansions ◽  
The Cost

Hardware double precision is often insufficient to solve large scientific problems accurately. Computing in higher precision defined by software causes significant computational overhead. The application of parallel algorithms compensates for this overhead. Newton's method to develop power series expansions of algebraic space curves is the use case for this application.

scholarly journals CONGRUENCES AMONG POWER SERIES COEFFICIENTS OF MODULAR FORMS

2013 ◽  
Vol 09 (06) ◽  
pp. 1447-1474
Author(s):  
RICHARD MOY
Keyword(s):  
Power Series ◽  
Modular Forms ◽  
Series Expansions ◽  
Modular Functions ◽  
Apéry Numbers ◽  
Congruence Relations ◽  
Power Series Expansions

Many authors have investigated the congruence relations among the coefficients of power series expansions of modular forms f in modular functions t. In a recent paper, R. Osburn and B. Sahu examine several power series expansions and prove that the coefficients exhibit congruence relations similar to the congruences satisfied by the Apéry numbers associated with the irrationality of ζ(3). We show that many of the examples of Osburn and Sahu are members of infinite families.

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