(∑_(n = 1)^∞) (1/((n^5) + 1))

N[-(1/5) RootSum[
          2 + 5 #1 + 10 #(1^2) + 10 #(1^3) + 5 #(1^4) + #(1^5) &, 
          PolyGamma[0, -#1]/(1 + 4 #1 + 6 #(1^2) + 4 #(1^3) + #(1^4)) &], 20]

Join[{{n, (Subscript[S, n]), 
            (Subscript[R, n])}}, 
Table[{m, (∑_(n = 1)^m) (1/((n^5) + 1)), 
            N[Abs[(∑_(n = 1)^m) (1/((n^5) + 1)) - (∑_(n = 1)^∞) (1/((n^5) + 1))], 20]}, {m, 1, 15}]] // 
      TableForm