Positive rational series for reciprocal powers of Catalan's constant and Dirichlet beta values

Let $\beta (s)={\sum }_{k=0}^{\mathrm{\infty }}(-1{)}^k(2k+1{)}^{-s}$ and let $G=\beta (2)$ be Catalan’s constant. We develop two families of positive series for reciprocal powers $\beta (s{)}^{-r}$. The first is obtained from the classical Euler transformation and is evaluated at $1\mathrm{/}2$; it is valid for real $s>0$. A second transformation, valid for $s\ge 2$, gives a faster series evaluated at $1\mathrm{/}3$. We derive finite-sum and integral formulas for the base coefficients, together with a positive recurrence and a composition formula for arbitrary reciprocal powers. When $s$ is an integer, all coefficients are rational. We also give explicit remainder estimates and determine the exact root-convergence rates of the two families. As applications, we obtain positive rational series for every reciprocal power of Catalan’s constant and for reciprocal powers of $\pi$ arising from odd values of the Dirichlet beta function.