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SerieEntiere - ` LES SERIES ENTIERES ET CONVERGENCE `...

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LES S ´ ERIES ENTI ` ERES ET CONVERGENCE D ´ EVELOPPEMENT DE FONCTIONS EN S ´ ERIES ENTI ` ERES ef´ erences MTH1101: S ´ ERIES ENTI ` ERES MOHAMMED SADDOUNE ´ Ecole Polytechnique de Montr´ eal, epartement de Math´ ematiques et G´ enie Industriel AUTOMNE 2009 MOHAMMED SADDOUNE MTH1101: S ´ ERIES ENTI ` ERES
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LES S ´ ERIES ENTI ` ERES ET CONVERGENCE D ´ EVELOPPEMENT DE FONCTIONS EN S ´ ERIES ENTI ` ERES ef´ erences 1 LES S ´ ERIES ENTI ` ERES ET CONVERGENCE Les s´ eries enti` eres Convergence Exercice 2 D ´ EVELOPPEMENT DE FONCTIONS EN S ´ ERIES ENTI ` ERES Substitution erivation et Int´ egration Exemples et Exercices 3 ef´ erences MOHAMMED SADDOUNE MTH1101: S ´ ERIES ENTI ` ERES
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LES S ´ ERIES ENTI ` ERES ET CONVERGENCE D ´ EVELOPPEMENT DE FONCTIONS EN S ´ ERIES ENTI ` ERES ef´ erences Les s´ eries enti` eres Convergence Exercice efinition d’une s´ erie enti` ere Une s´ erie enti` ere est une s´ erie de la forme P + n =0 c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + . . . o`u x est une variable et c n des constantes appel´ ees coefficients de la s´ erie. La somme de la s´ erie est f(x) = P + n =0 c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + . . . efinition d’une s´ erie enti` ere centr´ e en a Une s´ erie enti` ere est de la forme P + n =0 c n ( x - a ) n = c 0 + c 1 ( x - a ) + c 2 ( x - a ) 2 + c 3 ( x - a ) 3 + . . . est appel´ ee erie enti` ere en ( x - a ) ou simplement erie enti` ere centr´ ee en a . Remarque Quand x = a, tous les termes sont nuls pour n 1, ce qui rend la s´ erie enti` ere centr´ ee en a toujours convergente. MOHAMMED SADDOUNE MTH1101: S ´ ERIES ENTI ` ERES
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LES S ´ ERIES ENTI ` ERES ET CONVERGENCE D ´ EVELOPPEMENT DE FONCTIONS EN S ´ ERIES ENTI ` ERES ef´ erences Les s´ eries enti` eres Convergence Exercice Remarque On voit que la s´
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