Gosper's Algorithm

Gosper's algorithm is a procedure in symbolic computation for finding closed-form solutions to indefinite sums of hypergeometric terms, that is, sums whose consecutive-term ratio is a rational function, in cases where the resulting sum is itself hypergeometric. It solves for a closed form S(n) such that the sum of the terms from 1 to n equals S(n) minus S(0), operating in two main steps: first isolating a polynomial factor that simplifies the ratio between consecutive terms of the target sum, and then solving a system of linear equations to determine a polynomial that yields the desired closed form. It was developed by Bill Gosper in the 1970s while he was working on the Macsyma computer algebra system, first at the Stanford Artificial Intelligence Laboratory and later at MIT. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/