Computation of certain infinite series of the form ∑f(n)nk∑f(n)nk for arbitrary real-valued k

We provide a general method from which one may obtain more arbitrary closed form solutions for infinite series of the form ∑f(n)nk∑f(n)nk, for k∈Rk∈R and sufficiently well-behaved f(n)f(n). We illustrate the methods for the special case of f(n)=an/n!f(n)=an/n!, obtaining generalizations to the Bell numbers and polynomials.