  In this note we prove that for all $a \in \mathbb{N}$, $x \in \mathbb{R}_+
\cup \{0\}$, and $s \in \mathbb{C}$ with $\Re(s) > a + 2$, the (alternating)
weighted series of the Hurwitz zeta function, $$ \sum_{k \geq 1} (\pm 1)^k (k +
x)^a\zeta(s,k + x), $$ resolves into a finite combination of Hurwitz (Lerch)
zeta functions. This applies in Marichal and Zena\"idi's theory on analogues of
the Bohr-Mollerup theorem for higher-order convex functions.
