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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]], let $(f_k)$ and $(g_k)$ be sequences in $R$, and let $m \\le n$ be integers.\nWrite $\\Delta f_k = f_{k+1} - f_k$ and $\\Delta g_k = g_{k+1} - g_k$ for the [[forward differences|TXTR68V4R1]].\nThen\n$$\\sum_{k=m}^{n} f_k\\, \\Delta g_k = f_{n+1} g_{n+1} - f_m g_m - \\sum_{k=m}^{n} g_{k+1}\\, \\Delta f_k.$$\n\n## Remarks\n\nThe product rule for differences is $\\Delta(f g)_k = f_k \\Delta g_k + g_{k+1} \\Delta f_k$, which one checks by expanding.\nSumming it from $k = m$ to $n$, the left side telescopes to $f_{n+1} g_{n+1} - f_m g_m$, and this gives the formula.\nA common form, often called the Abel transformation, uses partial sums $A_k = a_1 + \\dots + a_k$:\n$$\\sum_{k=1}^{n} a_k b_k = A_n b_n - \\sum_{k=1}^{n-1} A_k (b_{k+1} - b_k).$$\nExample: with $f_k = k$ and $g_k = 2^k$, $\\Delta g_k = 2^k$ and the formula gives $\\sum_{k=0}^{n} k\\, 2^k = (n+1) 2^{n+1} - \\sum_{k=0}^{n} 2^{k+1} = (n - 1) 2^{n+1} + 2$.\nFor $n = 2$ both sides equal $10$.\n\n## Explanations\n\n### L1 human\n\nSummation by parts moves a difference from one factor of a product to the other, at the price of boundary terms.\nIt is the discrete form of [[integration by parts|08Y1BBQTK9]].\nIt helps when one factor is easy to sum and the other changes slowly.\n\n### L2 human\n\nThe Abel form is the key step in [[Dirichlet's test|0V9ME8GQE7]] and [[Abel's test|0CN97GADE5]] for the convergence of series $\\sum a_k b_k$.\nIn Dirichlet's test, bounded partial sums $A_k$ and monotone $b_k$ tending to $0$ make the right side converge.\nIts continuous-variable version is [[Abel's summation formula|V5EPT2PX9Z]], used in analytic number theory.\nIn the [[calculus of finite differences|FT9JQA58KJ]] it is proved from the [[fundamental theorem of the calculus of finite differences|MNSSV4QHD1]] applied to the product $f g$.\n",
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