The main contribution of this paper is to propose a closed expression for the Ramanujan constant alternating series, based on Euler–Boole summation formula. Such an not present in literature. We also highlight only choice parameter formula proposed by Hardy series positive terms, so value obtained as agrees with other methods divergent series. Additionally, we derive closed-formula chosen, unde...

and Applied Analysis 3 Proposition 8 (see [33]). The Bernoulli’s polynomial Bj(x) satisfies the following properties: (i) B 0 (x) = 1, B 1 (x) = x − 1/2, B k (x) = ∑ k j=0 C j k B j x , (ii) B 2k+1 (1) = B 2k+1 (0) = B 2k+1 = 0, (iii) B 2k (1) = B 2k (0) = B 2k , (iv) B k (x) = (1/(k + 1))B 󸀠 k+1 (x), k = 1, 2, . . . . 3.2. The Euler-Maclaurin Method and Discretization. Let q > 0 be an integer,...

We prove that any odd dimensional geometric graph G = (V,E) has zero curvature everywhere. To do so, we prove that for every injective function f on the vertex set V of a simple graph the index formula 1 2 [1 − χ(S(x))/2 − χ(Bf (x))] = (if (x) + i−f (x))/2 = jf (x) holds, where if (x) is a discrete analogue of the index of the gradient vector field ∇f and where Bf (x) is a graph defined by G an...

[This document is http://www.math.umn.edu/ ̃garrett/m/complex/notes 2014-15/09c Riemann and zeta.pdf] 1. Riemann’s explicit formula 2. Analytic continuation and functional equation of ζ(s) 3. Appendix: Perron identity [Riemann 1859] exhibited a precise relationship between primes and zeros of ζ(s). A similar idea applies to any zeta or L-function with analytic continuation, functional equation, ...

Using the Euler-Maclaurin summation formula the strictly increasing convergence lim m → ∞ m j=1 j m m = e e − 1 is demonstrated.

where / ( / ) is a differentiate function defined on the real number interval [a, b), and x may be a real or complex number with x*0 and x*l. Obviously, the case for x = l of (1.1) could be generally treated by means of the well-known Euler-Maclaurin summation formula. The object of this paper is to find a general summation formula for (1.1) that could be applied readily to some interesting spe...