We use an original summation technique to obtain finite and explicit sum formulas for some divergent series associated with the zeta and related functions. We show that the sums of such series are expressed in terms of Bernoulli numbers. By considering Bernoulli polynomials, we find the finite formulas for the sums of certain divergent series of Euler and some other infinite series involving zeta functions. The possible application of the summation technique and obtained formulas to the evaluation of the Riemann zeta function at odd positive integers is discussed.