Some properties of the remainders in certain series representations for the constant π

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.

Automated summary

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.