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\begin{document}

\begin{center}
$%
\begin{tabular}
[c]{lll}%
\textbf{U.S.T.H.B. 2013-2014 Semestre 1} & \hspace{3.75cm} & \textbf{Math 3} :
\textbf{Séries\vspace{-0.25cm}}\\
\textbf{Faculté de Mathématiques} &  & \textbf{2}$^{\mathbf{\grave{e}me}}%
$\textbf{\ Lic, ST, Sections D et L}%
\end{tabular}
\ \ \ $\textbf{\vspace{0.25cm}}

\underline{\textbf{Série d'exercices N}$%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$\textbf{\ 3 : Séries entières}}\textbf{\vspace{0.25cm}}
\end{center}

\noindent\underline{\textbf{Exercice 1}}\textbf{\ :} \label{E1}Déterminer le
rayon de convergence et la nature pour $x=\pm R$ des séries entières suivantes
:\textbf{\vspace*{-0.25cm}}%
\[%
\begin{array}
[c]{lllll}%
1)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{n}{4^{n}}x^{n} &  & 2)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
e^{-3n}x^{n} &  & 3)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\left(  \text{Log\thinspace}\left(  1+\dfrac{1}{2^{n}}\right)  \right)
x^{n}\\
4)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{x^{n}}{4^{n}\left(  2n+1\right)  } &  & 5)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{\left(  2n\right)  !}{n!n^{n}}x^{n} &  & 6)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{a^{n}}{n!}x^{n},a>0.
\end{array}
\]
\textbf{\vspace*{-0.25cm}}\noindent\underline{\textbf{Exercice 2}}\textbf{\ :}
Donner le domaine de convergence des séries entières suivantes
:\textbf{\vspace*{-0.25cm}}%
\[%
\begin{array}
[c]{lll}%
1)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{n}{4^{n}}x^{2n+1} &  & 2)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{n^{2}}{3^{n}+n}x^{3n-1}.
\end{array}
\]
\textbf{\vspace*{-0.25cm}}

\noindent\underline{\textbf{Exercice 3}}\textbf{\ :} Calculer le rayon de
convergence puis la somme des séries entières suivantes :\textbf{\vspace
*{-0.25cm}}%
\[%
\begin{array}
[c]{llll}%
1)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\left(  -1\right)  ^{n+1}nx^{2n+1} & 2)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{2n+3}{2n+1}x^{n} & 3)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{3n^{2}+1}{n!}x^{n} & 4)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{1}{n2^{n}}x^{n}.
\end{array}
\]
\textbf{\vspace*{-0.25cm}}

\noindent\underline{\textbf{Exercice 4}}\textbf{\ :} Donner le développement
en séries entières des fonctions suivantes et donner le domaine de
convergence.\textbf{\vspace*{-0.25cm}}%
\[%
\begin{array}
[c]{llll}%
1)\ f\left(  x\right)  =\dfrac{2}{x^{2}-4x+3} & 2)\ f\left(  x\right)
=\text{Log\thinspace}\left(  1-4x^{2}\right)  & 3)\ f\left(  x\right)
=\left(  1+x\right)  e^{-x} & 4)\ f\left(  x\right)  =\int_{0}^{x}e^{-t^{2}%
}dt.
\end{array}
\]


\begin{center}
\underline{\textbf{Exercices supplémentaires}}\textbf{\vspace{0.25cm}}
\end{center}

\noindent\underline{\textbf{Exercice 1}}\textbf{\ :} Déterminer une solution
développable en séries entières de l'équation différentielle%
\[
2xy^{\prime\prime}+y^{\prime}-y=0,
\]
puis calculer la somme de la série obtenue à l'aide de fonctions élémentaires.

\noindent\underline{\textbf{Exercice 2}}\textbf{\ :} Calculer les intégrales
suivantes en développant en séries entières les fonctions sous les signes
d'intégration.\textbf{\vspace*{-0.25cm}}%
\[%
\begin{array}
[c]{lllll}%
1)\
%TCIMACRO{\dint _{0}^{1}}%
%BeginExpansion
{\displaystyle\int_{0}^{1}}
%EndExpansion
\dfrac{\sin x}{x}dx\text{ à }10^{-5}\text{ près} &  & 2)\
%TCIMACRO{\dint _{0}^{1}}%
%BeginExpansion
{\displaystyle\int_{0}^{1}}
%EndExpansion
e^{-x^{2}}dx\text{ à }10^{-3}\text{ près} &  & 3)\
%TCIMACRO{\dint _{0}^{\frac{\pi}{4}}}%
%BeginExpansion
{\displaystyle\int_{0}^{\frac{\pi}{4}}}
%EndExpansion
\sin\left(  x^{2}\right)  dx\text{ à }10^{-3}\text{ près.}%
\end{array}
\]



\end{document}