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  "body": "\n## Statement\n\nLet $(a_n)_{n \\ge 1}$ be a sequence of complex numbers, and for real $t$ put $A(t) = \\sum_{1 \\le n \\le t} a_n$.\nLet $x \\ge 1$ be a real number, and let $\\varphi$ be a [[continuously differentiable|06SKCHDQT5]] complex-valued function on $[1, x]$.\nThen\n$$\\sum_{1 \\le n \\le x} a_n \\varphi(n) = A(x) \\varphi(x) - \\int_1^x A(t) \\varphi'(t) \\, dt.$$\n\n## Remarks\n\nThe function $A$ is constant on each interval $[k, k+1)$, so the integral splits as $\\sum_k A(k) (\\varphi(k+1) - \\varphi(k))$ plus a last piece up to $x$.\nRearranging this finite sum gives the left side; this is summation by parts.\nThe formula is the discrete analogue of [[integration by parts|08Y1BBQTK9]], and it is integration by parts for the Riemann–Stieltjes integral $\\int \\varphi \\, dA$.\nExample: with $a_n = 1$ and $\\varphi(t) = 1/t$, one has $A(t) = \\floor{t}$, and\n$$\\sum_{1 \\le n \\le x} \\frac{1}{n} = \\frac{\\floor{x}}{x} + \\int_1^x \\frac{\\floor{t}}{t^2} \\, dt.$$\nWriting $\\floor{t} = t - \\{t\\}$ shows that the harmonic numbers minus the logarithm converge.\n\n## Explanations\n\n### L1 human\n\nAbel's summation formula turns a weighted sum $\\sum a_n \\varphi(n)$ into an integral of the partial sums $A(t)$.\nIt is useful when the partial sums of $a_n$ are well understood but the terms themselves are irregular.\nIt plays the role of integration by parts for sums.\n\n### L2 human\n\nIn analytic number theory one often knows a counting function $A(x)$, such as the number of primes up to $x$, and wants sums like $\\sum_{p \\le x} 1/p$.\nAbel's formula transfers estimates for $A$ into estimates for such weighted sums.\nIt also extends the [[Riemann zeta function|1XWBY4N0VT]] to a meromorphic function on $\\operatorname{Re}(s) \u003e 0$ with a single pole at $s = 1$, by taking $a_n = 1$ and $\\varphi(t) = t^{-s}$.\nThe [[Dirichlet's test|0V9ME8GQE7]] and [[Abel's test|0CN97GADE5]] for series rest on the same summation by parts.\n",
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