  Using Euler transformation of series we relate values of Hurwitz zeta
function at integer and rational values of arguments to certain rapidly
converging series where some generalized harmonic numbers appear. The form of
these generalized harmonic numbers carries information about the values of the
arguments of Hurwitz function. In particular we prove: $\forall k\in
\mathbb{N}:$ $\zeta (k,1)\allowbreak =\allowbreak \frac{2^{k-1}}{2^{k-1}-1}%
\sum_{n=1}^{\infty }\frac{H_{n}^{(k-1)}}{n2^{n}},$ where $H_{n}^{(k)}$ are
defined below generalized harmonic numbers. Further we find generating function
of the numbers $\hat{\zeta}(k)=\sum_{j=1}^{\infty }(-1)^{j-1}/j^{k}. $
