10 Indefinite summation

The problem of indefinite summation is very similar to the problem of indefinite integration, in fact, we can somehow think of it as a discrete analogon to the integration problem. Whereas in integration we start out with a continuous function f(x) and want to determine another function g(x) such that in indefinite summation we are given a sequence (a_n)_n∈ℕ and we want to determine another sequence (s_n)_n∈ℕ₀ (in which the function symbol ∑ is eliminated) such that any partial sum of the corresponding series can be expressed as Of course we expect that the existence of algorithmic solutions for this indefinite summation problem will depend crucially on the class of functions that we take as input and possible output.