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Chapter: Sum Manipulation Methods

In practicing arithmetics, there are often situations, in which we want to manipulate sums in order to simplify them or to change their form into another that is closer to some goal. The goal might be to prove something or to estimate the magnitude of the sum. This chapter is dedicated to some basic methods of sum manipulation.

For further useful techniques of sum manipulation please refer to discrete calculus, especially to indefinite sums and the fundamental theorem of difference calculus.

Table of Contents

1. Definition: Sums
2. Proposition: Basic Rules of Manipulating Finite Sums
3. Proposition: Double Summation
4. Proposition: Rule of Combining Different Sets of Indices
5. Proposition: The General Perturbation Method
6. Proposition: Abelian Partial Summation Method
7. Section: Closed Formulas for Sums
8. Proposition: Product of Two Sums (Generalized Distributivity Rule)

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