= Cauchy-Hadamard theorem
{c}
{title2=$R^{-1}=\limsup_{n\to\infty}|a_n|^{1/n}$}
{wiki=Cauchy–Hadamard_theorem}
For the <power series> $\sum_{n\ge0}a_n(z-c)^n$, the <Cauchy-Hadamard theorem> gives
$$
\frac1R=\limsup_{n\to\infty}|a_n|^{1/n},
$$
with $R=\infty$ when the <limit superior> is zero, and $R=0$ when it is infinite. Away from the centre, the <root test> explains the formula: the <limit superior> of the term roots at displacement $z-c$ is $|z-c|/R$, interpreted as zero for $R=\infty$ and infinite for $R=0$. Values below one give <absolute convergence>; values above one imply that the terms fail to tend to zero. A value of one leaves the boundary question open. At the centre the series converges regardless of $R$.
For $a_{2n}=1$ and $a_{2n+1}=0$, the coefficient roots alternate between one and zero, so their ordinary <limit> does not exist. Their <limit superior> is nevertheless one, giving $R=1$ for $\sum_{n\ge0}z^{2n}=1/(1-z^2)$ inside the <unit disk>. A ratio of consecutive <coefficients> would be undefined at the zero coefficients; the root-limsup formula handles the gap directly.
In the first part of his 1892 study of <functions> given by <Taylor series>, Jacques Hadamard introduced an upper-limit formulation for <sequences> and used coefficient roots to determine the <radius of convergence>. Sections 4–6 separate finite positive, zero, and infinite root-growth cases; the later parts study singularities and continuation. See https://www.numdam.org/item/JMPA_1892_4_8__101_0.pdf[Hadamard, Essai sur l'étude des fonctions données par leur développement de Taylor, Journal de mathématiques pures et appliquées, fourth series 8 (1892), pp. 107–109]. This original text explains the coefficient-growth question without implying that the study of <power series> began there.
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