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  "body": "\n## Summary\n\nGosper's algorithm decides, for a hypergeometric term $t_k$, whether its partial sums $\\sum_{j \u003c k} t_j$ are a hypergeometric term plus a constant.\nWhen they are, it finds that term.\nThis is the discrete analogue of deciding whether an integral can be done in closed form.\nIt reduces the question to finding a polynomial solution of a linear recurrence with polynomial coefficients.\nIt is a building block of [[Zeilberger's algorithm|VA7MH05Z3B]] and of the search for [[WZ pairs|WC0QNABPZ4]].\n\n## Explanations\n\n### L1 human\n\nSome sums have a neat closed form, such as $\\sum_{j=0}^{k-1} j\\cdot j! = k! - 1$.\nOthers, such as the partial sums $1 + \\frac12 + \\cdots + \\frac1k$ of the harmonic series, have none of the same kind.\nGosper's algorithm looks for an answer $T_k$ that is the term $t_k$ times a rational function of $k$, with $T_{k+1} - T_k = t_k$.\nIt either finds such a $T_k$ or proves that none exists.\n",
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