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  "body": "\n## Summary\n\nA Padé approximant of a power series is a [[rational function|0WCZH4AFMT]] with prescribed degrees.\nIts own power series agrees with the given one to as high an order as these degrees allow.\nFor numerator degree at most $m$ and denominator degree at most $n$, the agreement is up to the power $z^{m+n}$.\nPadé approximants often approximate a function far beyond the disc of convergence of its Taylor series.\nThey can also show the location of poles.\nThey are closely linked to [[continued fractions|0BK6JBK58W]] and to orthogonal polynomials.\n\n## Explanations\n\n### L1 human\n\nA [[Taylor polynomial|0HFSFVMWQB]] approximates a function by a polynomial, which cannot have poles and tends to infinity far away.\nA Padé approximant uses a quotient of two polynomials instead, so it can imitate poles and limits at infinity.\nThe coefficients are chosen so that the quotient matches as many Taylor coefficients as possible.\nFor example, $(1 + z/2)/(1 - z/2)$ matches $e^z = 1 + z + z^2/2 + \\cdots$ up to the term $z^2$.\nThe approximants for all pairs of degrees form a table, the Padé table.\n",
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