Consider two series \(\sum a_n\) and \(\sum b_n\) of complex numbers, convergent to \(A\) and \(B\), respectively, and denote by \(A_n := \sum_{i=1}^n a_i\) and \(B_n := \sum_{i=1}^n b_i\) the sequence of their partial sums. Also, for positive integers \(n\) and \(r\), define the partial sums of the Riemann zeta function by: \[ \zeta_n(r) := \sum_{k=1}^n\frac{1}{k^r} \] Mainly, the paper investigates the sums
\[ F_n(A, B; x) := \sum_{k=1}^n (A-A_k)(B-B_k)x^k \] and
\[ F(A, B; \zeta(r)) := \sum_{n=1}^\infty (A-A_n)(B-B_n)(\zeta(r) - \zeta_n(r)) \] and provides explicit formulas for both. The paper also provides several other identities, relating the harmonic numbers to multiple zeta values. Some other series, involving multiple zeta star values, are evaluated, as well.