We derive the following globally convergent series for the Riemann zeta function and the Dirichlet beta function$$\zeta(s)=\frac{1}{2^{s}-2}\sum_{k=0}^{\infty}\frac{1}{2^{2k+1}}\binom{2k+1}{k+1}\sum_{m=0}^{k}\binom{k}{m}\frac{(-1)^{m}}{(m+1)^{s}} \qquad \mbox{(where $s \neq 1+\frac{2\pi i n}{\ln 2}$)},$$$$\beta(s)=\frac{1}{4^{s}}\sum_{k=0}^{\infty}\frac{1}{(k+1)!}\left(\left(\frac{3}{4}\right)^{(k+1)}-\left(\frac{1}{4}\right)^{(k+1)}\right)\sum_{m=0}^{k}\binom{k}{m}\frac{(-1)^{m}}{(m+1)^{s}}$$using a globally convergent series for the polylogarithm function, and integrals representing the Riemann zeta function and the Dirichlet beta function. To the best of our knowledge, these series representations are new. Additionally, we give another proof of Hasse's series representation for the Riemann zeta function.
A certain class of rapidly convergent series representations for ζ(2n+1)
