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  "body": "\n## Statement\n\nLet $R \u003e 0$ and let $n \\ge 1$ be an integer.\nLet $f$ be a [[meromorphic function|0ZZ7ZYDR8W]] on the disc $D = \\set{z \\in \\C : \\abs{z} \u003c R}$ that is analytic at $0$ and has exactly $n$ [[poles|066E7PJCYE]] in $D$, counted with multiplicity.\nLet $P$ be the set of these poles.\nLet $[m/n]_f$ be the [[Padé approximant|1F16G4V64X]] of $f$ at $0$ with numerator degree at most $m$ and denominator degree at most $n$.\nThen:\n\n1. as $m \\to \\infty$, $[m/n]_f$ converges to $f$ [[uniformly|08VZ57PT26]] on every compact subset of $D \\setminus P$;\n2. for $m$ large, $[m/n]_f$ has exactly $n$ poles counted with multiplicity, and as $m \\to \\infty$ they converge to the poles of $f$, each with its multiplicity.\n\n## Remarks\n\nThe idea of the proof: let $Q(z) = \\prod_{p \\in P} (z - p)$, so that $Qf$ is analytic on $D$.\nThe denominators $q_m$ of $[m/n]_f$ are determined by $n$ linear conditions on the Taylor coefficients of $f$ of high order.\nThe asymptotics of these coefficients are dominated by the poles in $P$, and one shows that $q_m \\to Q$ after normalization, at a geometric rate.\nThen $q_m f - p_m$ is small by construction, and dividing by $q_m$ gives the convergence away from $P$.\nExample: $f(z) = e^z / (1 - z)$ has exactly one pole in every disc of radius $R \u003e 1$.\nSo the column $[m/1]_f$ converges to $f$ on compact subsets of $\\C \\setminus \\set{1}$, and the pole of $[m/1]_f$ tends to $1$.\nThe hypothesis that the number of poles equals $n$ matters.\nWith fewer poles the approximants may have spurious poles, and with more poles a denominator of degree $n$ cannot capture them all.\nThe theorem was proved by Robert de Montessus de Ballore in 1902.\n\n## Explanations\n\n### L1 human\n\nA Taylor polynomial cannot see beyond the first pole of a function: it converges only in a disc that stops there.\nSuppose the function has exactly $n$ poles in a larger disc, and the approximations may have $n$ poles.\nThen the Padé approximants find those poles and converge in the whole larger disc.\nSo rational approximation extends the reach of power series past their singularities.\n\n### L2 human\n\nThis is the basic convergence theorem for columns of the [[Padé table|MDM8F48X18]], where the denominator degree is fixed and the numerator degree grows.\nIt gives a practical way to locate the poles of a function from its Taylor coefficients, as the limits of the poles of the approximants.\nFor diagonal sequences $[n/n]_f$, where both degrees grow, convergence is much harder; uniform convergence can fail, which is the subject of the [[Baker–Gammel–Wills conjecture|YPTNBRWRY7]].\n",
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