On One method for constructing a family of approximations of zeta constants by rational fractions

We present a new method for fast approximation of zeta constants, i.e., values $\zeta(n)$ of the Riemann zeta function, $n\ge2$, $n$ is an integer, by rational fractions. The method makes it possible to fast approximate zeta constants and certain combinations of successive values of zeta constants by rational fractions. By choosing values of coefficients involved in the combinations, one can control the convergence rate of the approximations and the computation complexity for the zeta constants.