One idea to solve abstract Cauchy problems is to apply the Laplace transform to the equation Au(t) + f(t) = u'(t), which leads to an equation where the unknown function only appears once. Now various characterizations of Laplace transforms yield solvability conditions of the problem. In this paper the same aim is reached by using the concept of moment sequences in Banach spaces. We extend Widder’s condition on scalar moment sequences to the Banach space case and give a characterization of finite Laplace transforms. Afterward we apply the notation of moment sequences to the abstract Cauchy problem.