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  "body": "\n## Summary\n\nLet $(x_i)_{i \\in I}$ be a family in a commutative topological group, indexed by any set $I$.\nIt has sum $s$ when the finite partial sums $\\sum_{i \\in F} x_i$ converge to $s$ as the finite set $F \\subseteq I$ grows.\nThis defines infinite sums with no order on the index set, and the sum does not depend on any rearrangement.\nIn a Hausdorff group a family has at most one sum.\nThis is the notion of summability used in Mathlib (`HasSum`), and it covers series in $\\R$, in Banach spaces and in $p$-adic numbers.\n\n## Explanations\n\n### L1 human\n\nTo add infinitely many elements of a group without choosing an order, look at all finite sub-collections and their sums.\nThe family has sum $s$ if these finite sums get as close to $s$ as you like once the sub-collection contains a large enough finite set.\nFor real numbers this is stronger than ordinary convergence of a series.\nFor example, $1 - \\tfrac12 + \\tfrac13 - \\cdots$ converges as a series, but as a family it has no sum, because one can pick finite sets of only positive terms with arbitrarily large sums.\n",
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