Efficient approximations of finite and infinite real alternating p-series

For n ∈ N and p ∈ R the nth partial sum of the alternating p-series, known also as alternating generalized harmonic number of order p, { Mathematical Formulas } is given in the form { Mathematical Formulas } where k, q ∈ N with k [n/2] are parameters, controlling the magnitude of the error term r∗q(k, n, p). The function Sq(k, n, p) consists of 2(k + 1) + q simple summands and r∗q(k, n, p) is estimated for q −p + 1, as { Mathematical Formulas } Additionally, for p ∈ R+ and k, q ∈ N, we have { Mathematical Formulas }