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\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{\ 2 : Suites et séries de fonctions}}\textbf{\vspace{0.25cm}}
\end{center}

\noindent\underline{\textbf{Exercice 1}}\textbf{\ : }\label{SF1}Étudier la
convergence simple et uniforme de la suite de fonctions\textbf{\vspace
*{-0.25cm}}%
\[
f_{n}\left(  x\right)  =\dfrac{\sin\left(  nx\right)  }{1+n^{2}x^{2}}%
,\ \ x\in\mathbb{R}.
\]


\noindent\underline{\textbf{Exercice 2}}\textbf{\ : }Étudier la convergence
simple des séries de fonctions suivantes :\textbf{\vspace*{-0.25cm}}%
\[%
\begin{array}
[c]{llll}%
1)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{e^{nx}}{3^{n}} & 2)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\left(  \text{Log\thinspace}x\right)  ^{n} & 3)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{1}{1+n^{2}x^{2}} & 4)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\left(  \dfrac{x^{n}}{n}-\dfrac{x^{n+1}}{n+1}\right)  .
\end{array}
\]


\noindent\underline{\textbf{Exercice 3}}\textbf{\ :} Montrer que la série de
fonctions $%
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{x^{2}}{\left(  1+x^{2}\right)  ^{n}}$ est simplement convergente sur
$\mathbb{R}$ mais qu'elle n'est pas uniformément convergente sur $\left[
-1,1\right]  $.

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

\noindent\underline{\textbf{Exercice 1}}\textbf{\ :} Étudier la convergence
simple et uniforme des suites de fonctions suivantes.\textbf{\vspace
*{-0.25cm}}%
\[%
\begin{array}
[c]{lll}%
\begin{array}
[c]{l}%
1)\ f_{n}\left(  x\right)  =nxe^{-nx^{2}},\ x\in\mathbb{R}\bigskip\\
2)\ f_{n}\left(  x\right)  =\cos\frac{nx}{n+1},\ \ x\in\left[  0,\frac{\pi}%
{2}\right]
\end{array}
&  & 3)\ f_{n}\left(  x\right)  =\left\{
\begin{array}
[c]{lll}%
1 &  & \text{si }0\leq\left\vert x\right\vert <\frac{1}{n}\\
\frac{n}{n-1}\left(  1-\left\vert x\right\vert \right)  &  & \text{si }%
\frac{1}{n}\leq\left\vert x\right\vert <1\\
0 &  & \text{si }\left\vert x\right\vert \geq0
\end{array}
\right.
\end{array}
\text{.}%
\]


\noindent\underline{\textbf{Exercice 2}}\textbf{\ :} On considère la suite de
fonctions $\left(  f_{n}\right)  _{n}$ définie sur $\mathbb{R}$ par
$f_{n}\left(  x\right)  =\cos\frac{x}{n}$.

\textbf{a)} Étudier la convergence simple et uniforme de la suite de fonctions
$\left(  f_{n}\right)  _{n}$.

\textbf{b)} Comparer $\lim\limits_{n\rightarrow+\infty}%
%TCIMACRO{\dint _{0}^{1}}%
%BeginExpansion
{\displaystyle\int_{0}^{1}}
%EndExpansion
f_{n}\left(  x\right)  dx$ et $%
%TCIMACRO{\dint _{0}^{1}}%
%BeginExpansion
{\displaystyle\int_{0}^{1}}
%EndExpansion
\lim\limits_{n\rightarrow+\infty}f_{n}\left(  x\right)  dx$.

\noindent\underline{\textbf{Exercice 3}}\textbf{\ :} Étudier la convergence
simple et uniforme des séries de fonctions suivantes :\textbf{\vspace
*{-0.25cm}}%
\[%
\begin{array}
[c]{llll}%
1)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{nx^{2}}{n^{3}+x^{2}} & 2)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{\sin\left(  nx\right)  }{n^{2}+x^{2}} & 3)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{nx}{e^{nx}} & 4)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{1}{\sqrt{n}}\left(  x^{2n}-x^{2n+1}\right)  .
\end{array}
\]


\noindent\underline{\textbf{Exercice 4}}\textbf{\ :} Calculer\textbf{\vspace
*{-0.25cm}}%
\[%
\begin{array}
[c]{lll}%
1)\ \lim\limits_{x\rightarrow0}%
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{1}{2^{n}n^{x}} &  & 2)\ \lim\limits_{x\overset{<}{\rightarrow}1}%
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{\left(  -1\right)  ^{n+1}x^{n}}{n\left(  1+x^{n}\right)  }.
\end{array}
\]



\end{document}