Rational zeta series
Specifically, given a real number x, the rational zeta series for x is given by where each qn is a rational number, the value m is held fixed, and ζ(s, m) is the Hurwitz zeta function.This last series follows from the general identity which in turn follows from the generating function for the Bernoulli numbers Adamchik and Srivastava give a similar series A number of additional relationships can be derived from the Taylor series for the polygamma function at z = 1, which is The above converges for |z| < 1.Many series involving the binomial coefficient may be derived: where ν is a complex number.Similar series may be obtained by simple algebra: and and and For integer n ≥ 0, the series can be written as the finite sum The above follows from the simple recursion relation Sn + Sn + 1 = ζ(n + 2).This process may be applied recursively to obtain finite series for general expressions of the form for positive integers m. Similar series may be obtained by exploring the Hurwitz zeta function at half-integer values.