The Sums of Alternating Series of Odd Powers of the Reciprocals of Odd Positive Integers

This paper is an extension of a recent work done by the author [4] and here the sums of alternating series of odd powers (up to fifteen) of the reciprocals of odd positive integers are computed. Following this method, the sum of the series of any higher power could be calculated. In the process of computing these sums, the sums of the series of even powers of reciprocals of odd positive integers have been reestablished and enabled the author to compute the values of Riemann’s zeta function for even positive integers.