Journal
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
Volume 116, Issue 3, Pages -Publisher
SPRINGER-VERLAG ITALIA SRL
DOI: 10.1007/s13398-022-01271-z
Keywords
Constant pi; Psi function; Polygamma functions; Asymptotic expansion; Inequality
Categories
Funding
- Key Science Research Project in Universities of Henan [20B110007]
- Fundamental Research Funds for the Universities of Henan Province [NSFRF210446]
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This paper examines the properties of the remainders in various series representations for the constant pi, including analytical representations, asymptotic expansions and inequalities. Specifically, the paper addresses a series summation problem and provides an integral representation.
The constant pi has many series representations. In this paper, we consider some properties of the remainders in certain series representations for the constant pi, including analytical representations, asymptotic expansions and inequalities. In 1963, Frame proposed the following problem: Sum the series S = Sigma(infinity)(K=0) (GRAPHICS) (-16)(-K) (2K + 1)(-2) Subsequently, Weinmann proved the required sum S = pi(2)/10 We here establish the following integral representation of S: S = 2 integral(1)(0) 1/x ln (x/2+root x(2)/4 +) dx = -2 integral(1)(0) In x/root x(2) +4 dx = pi(2)/10.
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