The following summation method can be used for sums of products and is due to Niels Henrik Abel (1802 - 1829).

Proposition: Abelian Partial Summation Method

Let $a_1,\ldots,a_n$ and $b_1,\ldots,b_n,b_{n+1}$ be some given elements of a unit ring $(R,+,\cdot).$ Then it is possible to reformulate the sum $\sum_{k=1}^n a_kb_k$ as follows: $$\sum_{k=1}^n a_kb_k = A_nb_{n+1}+\sum_{k=1}^n A_k(b_k-b_{k+1})$$ with $A_k:=\sum_{i=1}^k a_i.$

Proofs: 1

Proofs: 1
Propositions: 2

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  1. Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition