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  "body": "\n## Definition\n\nLet $D \\subseteq \\R$ and let $x_0$ be a limit point of $D$, possibly $\\pm\\infty$.\nA sequence of functions $\\varphi_0, \\varphi_1, \\varphi_2, \\ldots$ on $D$ is an *asymptotic sequence* as $x \\to x_0$ if $\\varphi_{n+1}(x) = o(\\varphi_n(x))$ as $x \\to x_0$ for every $n$.\nLet $f$ be a function on $D$ and let $a_0, a_1, \\ldots$ be numbers.\nThe formal series $\\sum_{n \\ge 0} a_n \\varphi_n$ is an *asymptotic expansion* of $f$ as $x \\to x_0$, written\n$$f(x) \\sim \\sum_{n=0}^{\\infty} a_n \\varphi_n(x) \\quad (x \\to x_0),$$\nif for every $N \\ge 0$,\n$$f(x) - \\sum_{n=0}^{N} a_n \\varphi_n(x) = o(\\varphi_N(x)) \\quad (x \\to x_0).$$\n*Asymptotic analysis* is the study of such expansions and of [[asymptotic equivalence|0QR92XBV6H]] $f \\sim g$, meaning $f / g \\to 1$.\nIts estimates are written with [[big O|0TGV80E6A4]] and [[little o|00T5EPFQB9]] notation.\n\n## Remarks\n\nExample: $f(x) = 1 / (x - 1)$ on $(1, \\infty)$ as $x \\to \\infty$, with $\\varphi_n(x) = x^{-n-1}$.\nHere $f(x) - \\sum_{n=0}^{N} x^{-n-1} = x^{-N-1} / (x - 1) = o(x^{-N-1})$, so $f(x) \\sim \\sum_{n \\ge 0} x^{-n-1}$.\n\nA divergent example: as $x \\to 0^+$, $\\int_0^\\infty \\frac{e^{-t}}{1 + x t} \\, dt \\sim \\sum_{n \\ge 0} (-1)^n n! \\, x^n$, although the series converges for no $x \\neq 0$.\n\nNon-example: $\\varphi_n(x) = x^n$ is not an asymptotic sequence as $x \\to \\infty$, since $x^{n+1}$ is not $o(x^n)$ there; it is one as $x \\to 0$.\n\nGiven the asymptotic sequence, the coefficients $a_n$ are unique.\nDifferent functions can have the same expansion: $e^{-x}$ has all coefficients $0$ with respect to $x^{-n}$ as $x \\to \\infty$.\n\n## Explanations\n\n### L1 human\n\nAn asymptotic expansion is a list of terms of decreasing size.\nStopping after any term leaves an error smaller than the last term kept, when $x$ is close enough to the limit point.\n",
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