Rational zeta series for $\zeta(2n)$ and $\zeta(2n+1)$

In this paper, we first find zeta series for $\zeta(2n)$ in terms of $\zeta(2k+1)$ and $\beta(2k)$, the Dirichlet beta function, using integration by parts and the power series representations for $\cot(x)$ and $\log(\sin(x))$. Further, we remark on some fast convergences for a few of these $\zeta(2k+1)$ terms, namely $\zeta(3)$, $\zeta(5)$, and $\zeta(7)$. After, we generalize these series for an arbitrary number of monomials on the denominator. In the second half of the paper, we do the same analysis but with different functions, $\psi(x)$ and $\log\Gamma(x)$. In the end, we extract rational zeta series with $\zeta(2n+1)$ on the numerator rather than $\zeta(2n)$. We again generalize these series for an arbitrary number of monomials on the denominator.