New upper bounds for Mathieu-type series (Q967139)

Mathieu's series is defined by \[ S(r)=\sum_{n=1}^{\infty}\frac{2n}{(n^2+r^2)^2}. \] The corresponding alternating series is \[ \tilde{S}(r)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{2n}{(n^2+r^2)^2}. \] The authors obtain upper bounds for the functions \(S(r)\) and \(\tilde{S}(r)\).