scholarly journals New Inequalities of Cusa–Huygens Type

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2101
Author(s):  
Ling Zhu
Keyword(s):  
Power Series ◽  
Bernoulli Numbers ◽  
Series Expansions ◽  
Power Series Expansions ◽  
New Inequalities ◽  
Better Than

Using the power series expansions of the functions cotx,1/sinx and 1/sin2x, and the estimate of the ratio of two adjacent even-indexed Bernoulli numbers, we improve Cusa–Huygens inequality in two directions on 0,π/2. Our results are much better than those in the existing literature.

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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