Rayon de convergence

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Le rayon de convergence d'une série entière est le nombre réel positif ou $+\infty$ égal à la borne supérieure de l'ensemble des modules des nombres complexes où la série converge (au sens classique de la convergence simple):

R=\sup \left\{|z|:z\in \mathbb {C} ,\sum a_{n}z^{n}{\text{ converge simplement }}\right\}\in \,[0,+\infty ]={\overline {\mathbb {R} ^{+}}}.

Propriétés

Si $R$ est le rayon de convergence d'une série entière, alors la série est absolument convergente sur le disque ouvert $D(0, R)$ de centre $0$ et de rayon $R$ . Ce disque est appelé disque de convergence. Cette convergence absolue entraine ce qui est parfois qualifié de convergence inconditionnelle : la valeur de la somme en tout point de ce disque ne dépend pas de l'ordre des termes. Par exemple, on a :

$\sum _{n=0}^{\infty }{a_{n}z^{n}}=\sum _{n=0}^{\infty }a_{2n}z^{2n}+\sum _{n=0}^{\infty }a_{2n+1}z^{2n+1}$ ;
$\sum _{n=0}^{\infty }\sum _{k=0}^{\infty }{a_{n}b_{k}z^{n+k}}=\left(\sum _{n=0}^{\infty }a_{n}z^{n}\right)\left(\sum _{k=0}^{\infty }b_{k}z^{k}\right)\ \ \forall |z|<\min(R_{1},R_{2})$ , où $R_{1}$ et $R_{2}$ sont les rayons de convergence des deux séries entières (voir Produit de Cauchy).

Si la série entière $\sum _{n=0}^{\infty }{a_{n}z^{n}}$ a pour rayon de convergence $R$ , alors :

la convergence est même normale (donc uniforme) sur tout compact inclus dans $D(0, R)$ ;
pour tout complexe $z$ tel que $| z | > R$ , la série diverge grossièrement ;
pour tout complexe $z$ tel que $| z | = R$ , la série peut soit diverger, soit converger ;
l'inverse du rayon $R$ est donné par le théorème de Cauchy-Hadamard : ${\frac {1}{R}}=\limsup _{n\to \infty }{\sqrt[{n}]{|a_{n}|}}\leq \limsup _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|$ , où $lim sup$ désigne la limite supérieure ;
si $R$ est non nul, alors la somme $f$ de la série entière est une fonction holomorphe sur $D(0, R)$ , où l'on a $f^{(k)}(z)=\sum _{n=k}^{\infty }{\frac {n!}{(n-k)!}}a_{n}z^{n-k}$ ;
si le rayon $R$ est infini, alors la série entière est appelée fonction entière.

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