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  "body": "\n## Definition\n\nLet $D$ be a subset of $\\R$ or of $\\C$, and let $x_0$ be a limit point of $D$, possibly $\\infty$.\nAn *asymptotic sequence* at $x_0$ is a sequence of functions $\\varphi_0, \\varphi_1, \\varphi_2, \\ldots$ on $D$ such that\n$$\\varphi_{n+1}(x) = o(\\varphi_n(x)) \\quad \\text{as } x \\to x_0 \\text{ in } D$$\nfor each $n \\ge 0$, where $o$ is [[little o notation|04TSHB9H8D]].\nLet $f$ be a function on $D$ and let $a_0, a_1, a_2, \\ldots$ be complex numbers.\nThe formal series $\\sum_{n \\ge 0} a_n \\varphi_n$ is an *asymptotic expansion* of $f$ at $x_0$ if for each $N \\ge 0$\n$$f(x) - \\sum_{n=0}^{N} a_n \\varphi_n(x) = o(\\varphi_N(x)) \\quad \\text{as } x \\to x_0 \\text{ in } D.$$\nThis is written $f(x) \\sim \\sum_{n \\ge 0} a_n \\varphi_n(x)$ as $x \\to x_0$.\n\n## Remarks\n\nExample: on $D = (1, \\infty)$ with $x_0 = \\infty$ and $\\varphi_n(x) = x^{-n-1}$, one has $\\frac{1}{x-1} \\sim \\sum_{n \\ge 0} x^{-n-1}$.\nCheck: $\\frac{1}{x-1} - \\sum_{n=0}^{N} x^{-n-1} = \\frac{x^{-N-1}}{x-1}$, and this is $o(x^{-N-1})$ because $\\frac{1}{x-1} \\to 0$.\nNon-example: $\\varphi_0(x) = x^{-1}$, $\\varphi_1(x) = x^{-1}$ is not an asymptotic sequence at $\\infty$, because $\\varphi_1 = o(\\varphi_0)$ fails.\nFor a fixed asymptotic sequence, the coefficients are unique: $a_N$ is the limit of $\\big(f(x) - \\sum_{n\u003cN} a_n \\varphi_n(x)\\big)/\\varphi_N(x)$ as $x \\to x_0$, whenever $\\varphi_N$ does not vanish near $x_0$.\nDifferent functions can have the same expansion: $e^{-x}$ and $0$ both have the zero expansion in powers of $x^{-1}$ as $x \\to +\\infty$.\nThe most common case is the power scale $\\varphi_n(x) = x^{-n}$ at $\\infty$ or $\\varphi_n(z) = z^n$ at $0$; an equivalent condition is then that the error after $N$ terms is $O(x^{-N-1})$, respectively $O(z^{N+1})$.\n",
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