Evaluation of convergent series by using finite parts

We present a method to find the sum of a convergent series based on the computation of Hadamard finite part limits of partial sums. We give several illustrations, the main being the formulas for convergent series of the type ∑ ∞ n = 2 ( − 1 ) n ζ ( n , a ) b n + k n + k , where ζ ( s , a ) is Hurwitz zeta function, | b | ≤ | a | , b ≠ − a , and k ∈ N .