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\begin{document}

\begin{center}
$\vspace{-0.3cm}$

$%
\begin{tabular}
[c]{lll}%
\textbf{U.S.T.H.B. 2013-2014 Semestre 1} & \hspace{3.75cm} & \textbf{Math 3} :
\textbf{Séries\vspace{-0.25cm}}\\
\textbf{Faculté de Mathématiques} &  & \textbf{2}$^{\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{\ 1 : Séries numériques}}\textbf{\vspace{0.25cm}}
\end{center}

\noindent\underline{\textbf{Exercice 1}}\textbf{\ : }\label{SN1}Soient les
deux suites $\left(  u_{n}\right)  $ et $\left(  v_{n}\right)  $, on suppose
que pour tout $n$ dans $\mathbb{N}$\textbf{\vspace*{-0.25cm}}%
\[
u_{n}=v_{n+1}-v_{n}.
\]
\textbf{a)} Montrer que la suite $\left(  v_{n}\right)  $ et la série $%
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
u_{n}$ sont de même nature.\newline\textbf{b)} \underline{\textit{Application
:}} Montrer que la série suivante est convergente et calculer sa somme.%
\[%
%TCIMACRO{\dsum \limits_{n=2}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=2}^{+\infty}}
%EndExpansion
\text{Log}\left(  1-\dfrac{1}{n^{2}}\right)
\]


\noindent\underline{\textbf{Exercice 2}}\textbf{\ : }Étudier la convergence
des séries numériques suivantes en calculant leur somme.%
\[%
\begin{array}
[c]{lllll}%
1)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{2^{n+1}}{3^{n}} & 2)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{1}{n\left(  n+1\right)  } & 3)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{2}{n\left(  n+2\right)  } & 4)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{2n+1}{n^{2}\left(  n+1\right)  ^{2}} & 5)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\text{Log}\dfrac{n\left(  n+2\right)  }{\left(  n+1\right)  ^{2}}.
\end{array}
\]


\noindent\underline{\textbf{Exercice 3}}\textbf{\ :} Établir la divergence des
séries suivantes :%
\[%
\begin{array}
[c]{lllllll}%
1)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\cos\dfrac{1}{\sqrt{n}} &  & 2)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
n! &  & 3)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\left(  -1\right)  ^{n} &  & 4)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\left(  \sqrt{n+1}-\sqrt{n}\right)  .
\end{array}
\]
\noindent\underline{\textbf{Exercice 4}}\textbf{\ :} Trouver la nature de la
série $%
%TCIMACRO{\dsum }%
%BeginExpansion
{\displaystyle\sum}
%EndExpansion
u_{n}$ qui a pour terme général $u_{n}$ en utilisant le critère indiqué.

\noindent\textbf{a)} \underline{\textit{Critère de comparaison}} $%
\begin{array}
[c]{llll}%
1)\ u_{n}=\dfrac{1}{\sqrt{n+1}} & 2)\ u_{n}=\sin\dfrac{1}{n^{\alpha}}%
,\alpha>0 & 3)\ u_{n}=e^{\frac{1}{n}}-1 & 4)\ u_{n}=\text{Log}\left(
1+\frac{1}{2^{n}}\right)  .
\end{array}
\medskip$

\noindent\textbf{b)} \underline{\textit{Critère de D'Alembert}} $%
\begin{array}
[c]{llll}%
1)\ u_{n}=\dfrac{1\times3\times...\left(  2n-1\right)  }{n!3^{n}} &
2)\ u_{n}=\dfrac{n^{n}}{n!} & 3)\ u_{n}=n^{3}\sin\dfrac{\pi}{3^{n}} &
4)\ u_{n}=\dfrac{a^{n}}{n^{n}},a>0.
\end{array}
\medskip$

\noindent\textbf{c)} \underline{\textit{Critère de Cauchy}} $%
\begin{array}
[c]{lll}%
1)\ u_{n}=\left(  \dfrac{n}{n+1}\right)  ^{n^{2}} & 2)\ u_{n}=2^{-n}\left(
\dfrac{n+1}{n}\right)  ^{n^{2}} & 3)\ u_{n}=\left(  \dfrac{n+1}{2n-1}\right)
^{2n}a^{n},a>0.
\end{array}
$

\noindent\underline{\textbf{Exercice 5}}\textbf{\ :} Étudier les séries
numériques suivantes :%
\[%
\begin{array}
[c]{llll}%
1)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{a^{n}}{n^{\alpha}},a>0,\alpha\in\mathbb{R} & 2)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{3^{n}+n}{a^{n}},a>0 & 3)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
n^{\alpha}e^{-\sqrt{n}},\alpha\in\mathbb{R} & 4)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{n^{2}+1}{n^{4}+n}\medskip\\
5)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\left(  1+\dfrac{a}{n}\right)  ^{-n^{2}},a>0 & 6)\
%TCIMACRO{\dsum \limits_{n=0}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=0}^{+\infty}}
%EndExpansion
\dfrac{\left(  n!\right)  ^{2}}{\left(  2n\right)  !} & 7)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{\text{Log\thinspace}n}{n^{\alpha}},\alpha>0 & 8)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
a^{\text{Log\thinspace}n},a>0.
\end{array}
\]


\noindent\underline{\textbf{Exercice 6}}\textbf{\ :} Étudier la convergence
absolue et la semi-convergence des séries suivantes :%
\[%
\begin{array}
[c]{lllll}%
1)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{\sin\left(  \alpha n\right)  }{n^{2}},\alpha\in\mathbb{R} & 2)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{\left(  -1\right)  ^{n}}{n^{2}+\text{Log\thinspace}n} & 3)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\dfrac{\cos^{2}n}{n} & 4)\
%TCIMACRO{\dsum \limits_{n=1}^{+\infty}}%
%BeginExpansion
{\displaystyle\sum\limits_{n=1}^{+\infty}}
%EndExpansion
\sin\left(  n\pi+\dfrac{1}{n}\right)  . &
\end{array}
\]



\end{document}