SumS[1, m] is the harmonic number S_ 1(m) = \sum_ {i=1}^m i^{-1}.
SumS[1,1,m] is \sum_{i=1}^m S_
1 (i)/i.
SumS[k,l,m] is \sum_{i=1}^m
S_l (i)/i^k.
SumS[r, n] represents
Sum[Sign[r]^i/i^Abs[r], {i, 1, n}].
SumS[r,s, n] is
Sum[Sign[r]^k/k^Abs[r] Sign[s]^j/j^Abs[s], {k, 1, n}, {j, 1, k}]
etc.
SumS[1, m - 1]S_1(m-1)
SumS[2, m - 1]S_2(m-1)
SumS[-1, m]S_{-1}(m)
SumS[1, m, Reduce -> True]S_1(m-1)+\frac{1}{m}
SumS[3, m + 2, Reduce -> True]S_3(m+1)+\frac{1}{(m+2)^3}
SetOptions[SumS, Reduce -> True];
SumS[3, m + 2]\frac{1}{m^3}+S_3(m-1)+\frac{1}{(m+1)^3}+\frac{1}{(m+2)^3}
SetOptions[SumS, Reduce -> False];
SumS[1, 4]\frac{25}{12}
SumS[1, 2, m - 1]S_{12}(m-1)
SumS[1, 1, 1, 11]\frac{31276937512951}{4260000729600}
SumS[-1, 4]-\frac{7}{12}
SumT[1, 4]-\frac{7}{12}