summationSimply

summationSimply is a formal framework and software tool for automated symbolic summation and simplification of series, designed to produce closed-form expressions where possible and to document the conditions of validity for finite and certain infinite sums. It emphasizes readability and reproducibility.

The concept builds on symbolic computation techniques, pattern recognition of known identities, and algebraic manipulation such as index shifts and telescoping. It aims to provide transparent derivations suitable for teaching and research.

Implementation and scope: The engine accepts expressions of the form sum_{k=a}^{b} f(k) and applies a suite of transformation rules, including changing the summation index, splitting sums, telescoping, partial fraction decomposition, and identification of common identities. It handles polynomials, rational functions, exponentials, and many special functions, and applies convergence checks for infinite sums.

Example and usage: Example: sum_{k=1}^{n} k = n(n+1)/2, which summationSimply derives by recognizing f(k)=k and applying a standard identity. Another example: sum_{k=1}^{n} (1/k - 1/(k+1)) telescopes to 1 - 1/(n+1). The tool favors stepwise, readable proofs.

Applications and limitations: Used in mathematics education, research, and symbolic computation workflows. Limitations include handling non-elementary sums or ambiguous inputs; results depend on specified domains and convergence conditions.

See also: Summation, Telescoping, Computer algebra system, Symbolic computation.