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  "body": "\n## Definition\n\nLet $f \\colon \\C \\to \\C$ be a function and let $a \\in \\C$.\nFor $k \\in \\N$ let $\\Delta^k f(a)$ be the [[$k$-th forward difference|E38801KVJ1]] of $f$ at $a$ with step $1$.\nFor $s \\in \\C$ let $\\binom{s}{k} = \\frac{s(s-1)\\cdots(s-k+1)}{k!}$, with $\\binom{s}{0} = 1$.\nThe *Newton series* of $f$ at $a$ is the [[series|0JH4MN4K1C]]\n$$\\sum_{k=0}^{\\infty} \\binom{x - a}{k} \\Delta^k f(a),$$\nviewed as a series of functions of $x \\in \\C$.\n\n## Remarks\n\nExample: for $f(x) = 2^x$ and $a = 0$, $\\Delta^k f(0) = 1$ for all $k$, so the Newton series is $\\sum_k \\binom{x}{k}$.\nThis is the binomial series of $(1 + z)^x$ at $z = 1$, which converges to $2^x$ for real $x \u003e -1$.\nNon-example: $\\sum_k \\frac{x^k}{k!}$ is the Taylor series of $e^x$ at $0$, not its Newton series.\nThe Newton series of $e^x$ at $0$ has coefficients $\\Delta^k e^x |_{x=0} = (e - 1)^k$, and since $e - 1 \u003e 1$ it diverges at every real $x$ that is not a natural number.\nAt $x = a + n$ with $n \\in \\N$, only the terms with $k \\le n$ are nonzero, and the sum is $f(a + n)$ by [[Newton's forward difference formula|DDHCMKHD2T]].\n",
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