\begin{enumerate}
  \item Let $a$ and $b$ be two real numbers with $a > 0$. Choose without justification the correct expression for $a ^ { b }$:\\
$(A) : \mathrm { e } ^ { b \ln ( a ) }$\\
$(B) : \mathrm { e } ^ { a \ln ( b ) }$\\
$(C) : \mathrm { e } ^ { \ln ( a ) \ln ( b ) }$.
  \item Let $x$ and $y$ be two real numbers such that $x < y$ and $t$ a real number in $]0,1[$. Compare $t ^ { x }$ and $t ^ { y }$.
  \item Give, without proof, the power series expansion of the real exponential function and give its domain of validity.
  \item We consider the function $\Gamma$ defined by $\Gamma ( x ) = \int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - t } \mathrm {~d} t$. We admit that this function is defined on $]0 , + \infty[$ and that, for all strictly positive real $x$:
$$\Gamma ( x + 1 ) = x \Gamma ( x )$$
Calculate $\Gamma ( 1 )$ and deduce, by using a proof by induction, the value of $\Gamma ( n + 1 )$ for $n \in \mathbb { N }$.
  \item For $x \in \mathbb { R }$, we denote, when it makes sense:
$$F ( x ) = \int _ { 0 } ^ { 1 } t ^ { t ^ { x } } \mathrm { d } t$$
where, as is customary, $t ^ { t ^ { x } } = t^{(t^{x})}$.
  \begin{enumerate}
    \item[5.1.] Determine the domain of definition of $F$.
    \item[5.2.] Determine the monotonicity of $F$.
    \item[5.3.] Prove that for all non-negative real $x$, we have: $F ( x ) \geqslant \frac { 1 } { 2 }$.
    \item[5.4.] Prove that $F$ is continuous on its domain of definition.
    \item[5.5.] Determine $\lim _ { x \rightarrow + \infty } F ( x )$ and $\lim _ { x \rightarrow - \infty } F ( x )$. The theorems used will be cited with precision and we will ensure that their hypotheses are well verified.
    \item[5.6.] Then carefully draw up the table of variations of $F$ and give a general sketch of its representative curve in an orthonormal coordinate system. We will admit that $F ^ { \prime } ( 0 ) = \frac { 1 } { 4 }$ and we will draw the tangent line at the point with abscissa $x = 0$.
  \end{enumerate}
  \item Let $x$ be a strictly positive real number. For every natural number $n$, we denote by $g _ { n }$ the function defined on $]0,1]$ by $g _ { n } ( t ) = \frac { t ^ { n x } \ln ^ { n } ( t ) } { n ! }$.
  \begin{enumerate}
    \item[6.1.] Prove that the series of functions $\sum _ { n \in \mathbb { N } } g _ { n }$ converges pointwise on $]0,1]$ and give its sum.
    \item[6.2.] Prove that, for every natural number $n , \int _ { 0 } ^ { 1 } \left| g _ { n } ( t ) \right| \mathrm { d } t = \frac { 1 } { n ! } \frac { \Gamma ( n + 1 ) } { ( n x + 1 ) ^ { n + 1 } }$.
    \item[6.3.] Finally establish that we have:
$$F ( x ) = \sum _ { n = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { n } } { ( 1 + n x ) ^ { n + 1 } }$$
  \end{enumerate}
\end{enumerate}