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  "body": "\n## Summary\n\nThe Newton series of a function $f$ at a point $a$ is the series $\\sum_{k \\ge 0} \\binom{x-a}{k} \\Delta^k f(a)$, built from the forward differences of $f$ at $a$.\nIt is the difference analogue of the Taylor series, with generalized binomial coefficients in place of powers.\nFor a polynomial it is a finite sum equal to the polynomial; for other functions it may or may not converge to $f$.\n\n## Explanations\n\n### L1 human\n\nA Taylor series rebuilds a function from its derivatives at one point.\nA Newton series tries to rebuild it from its differences at one point, that is, from its values at $a, a+1, a+2, \\dots$.\nAt the integers $x = a + n$ the series stops after $n + 1$ terms and gives exactly $f(a + n)$.\nBetween the integers it is an interpolation, which can work well or fail: the function $\\sin(\\pi x)$ vanishes at every integer, so its Newton series at $0$ is the zero series.\n",
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