Beside a short account on the summation/integration methods for slowly convergent series $($Laplace transform method for numerical and trigonometric series and the method of contour integration over a rectangle$)$, a method based on a new transformation of the series to a weighted integral, with respect to the weight function \[t\mapsto w(t)=\frac{1}{qrt{t}\cosh^2\dfrac{iqrt{t}}{2}}\] over $\RR_+$, is presented. Such a method is applied to calculation of the values of the Riemann zeta function. Several numerical examples are included.