Analysis of divergent series by Euler–Maclaurin summation

In this presentation, we will consider expressions of the form S(δ) = s=0 (s+ x)nE(s, δ), n ∈ N, (1) where x is a fixed real parameter, and the function E is such that the series converges for δ> 0, but E(s, 0) = E0 (a constant). Series of the this type are of great importance in the study of wave scattering by periodic structures [1, 2], and it is often necessary to determine the nature of the singularity of the function S(δ) at the point δ = 0. For the case where E(s, δ) = e−sδ, Nørlund [3, p. 53] used Euler–Maclaurin summation [4] to obtain lim δ→0+ s=0 (s+ x)ne−(s+x)δ −