Energy bounds for a class of singular potentials and some related series

Perturbation expansions up to third order for the generalized spiked harmonic oscillator Hamiltonians H = -d^2/dx^2+ x^2 + A/x^2 + lambda/x^alpha, A >= 0, 2gamma > alpha, gamma=1+(1/2)sqrt(1+4A), and small values of the coupling lambda > 0, are developed. Upper and lower bounds for the eigenvalues are computed by means of the procedure of Burrows et al [J. Phys. A: Math. Gen. 20, 889-897 (1987)] for assessing the accuracy of a truncated perturbation expansion. Closed-form sums for some related perturbation double infinite series then immediately follow as a result of this investigation.