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\begin{center}
$%
\begin{tabular}
[c]{lll}%
\textbf{USTHB 2014-2015 Semestre 2} & \hspace{3.75cm} & \textbf{Math 4} :
\textbf{Analyse complexe\vspace{-0.25cm}}\\
\textbf{Faculté de Mathématiques} &  & \textbf{2}$^{\mathbf{\grave{e}me}}%
$\textbf{\ Lic, ST-GP, Section G}%
\end{tabular}
\ \ \ \ $\textbf{\vspace{0.25cm}}

\underline{\textbf{Série d'exercices n}$%
%TCIMACRO{\U{b0}}%
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{{}^\circ}%
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$\textbf{\ 4 : Séries infinies,} \textbf{séries de Taylor, séries de Laurent}}
\end{center}

\noindent\underline{\textbf{Exercice 1}}\textbf{\ : \label{key6}}

\noindent Déterminer le domaine de convergence des séries$\bigskip$

$\hspace{0.5cm}$\textbf{a)} $%
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{\left(  -1\right)  ^{n-1}z^{2n-1}}{\left(  2n-1\right)  !}.\bigskip$

$\hspace{0.5cm}$b) $%
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
n!z^{n}.\vspace{1cm}$

\noindent\underline{\textbf{Exercice 2}}\textbf{\ :}

\noindent Soit $f\left(  z\right)  =$Log$\,\left(  1+z\right)  $, où l'on
considère la branche qui prend la valeur zéro pour $z=0$.

\textbf{a)} Développer $f\left(  z\right)  $ en série de Taylor au voisinage
de $z=0$.$\bigskip$

\textbf{b)} Déterminer le domaine de convergence de la série de \textbf{(a)}%
.$\bigskip$

\noindent\underline{\textbf{Exercice 3}}\textbf{\ :}

\noindent Déterminer le développement en série de Laurent des fonctions
suivantes au voisinage des singularités indiquées.

a) $f\left(  z\right)  =\dfrac{e^{2z}}{\left(  z-1\right)  ^{3}}%
;\ z=1$.$\bigskip$

\textbf{b)} $f\left(  z\right)  =\dfrac{z}{\left(  z+1\right)  \left(
z+2\right)  };\ z=-2$.


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