SumT[1, m] is the alternative harmonic number \sum _{i=1}^m (-1){}^{\wedge}i/i
SumT[r, n] represents
Sum[(-1)^i/i^r, {i,1,n}]
SumT[r,s, n] is
Sum[1/k^r (-1)^j/j^s, {k, 1, n}, {j, 1, k}].
SumT[1, m - 1]\tilde{S}_1(m-1)
SumT[2, m - 1]\tilde{S}_2(m-1)
SumT[1, m]\tilde{S}_1(m)
SumT[1, m, Reduce -> True]\tilde{S}_1(m-1)+\frac{(-1)^m}{m}
SumT[1, 4]-\frac{7}{12}
SumT[1, 2, m - 1]\tilde{S}_{12}(m-1)
SumT[1, 2, 42]-\frac{38987958697055013360489864298703621429610152138683927}{10512121660702378405316004964483761080879190528000000}
SumT[1, 4]-\frac{7}{12}
SumS[-1, 4]-\frac{7}{12}
SumT[1, 2, 12]-\frac{57561743656913}{21300003648000}
SumS[1, -2, 42]-\frac{38987958697055013360489864298703621429610152138683927}{10512121660702378405316004964483761080879190528000000}
Array[SumT, 6]\left\{-1,-\frac{5}{8},-\frac{179}{216},-\frac{1207}{1728},-\frac{170603}{216000},-\frac{155903}{216000}\right\}
Array[SumS[-2, 1, #1] &, 6]\left\{-1,-\frac{5}{8},-\frac{179}{216},-\frac{1207}{1728},-\frac{170603}{216000},-\frac{155903}{216000}\right\}