 Then equivalently, formula_31 is convergent, which implies that formula_32 and formula_33 converge by termwise comparison of non-negative terms.It suffices to show that the convergence of these series implies the convergence of formula_34 and formula_35, for then, the convergence of formula_36 would follow, by the definition of the convergence of complex-valued series.The preceding discussion shows that we need only prove that convergence of formula_37 implies the convergence of formula_38.