For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. The five questions of this exercise are independent.
We consider the script written in Python language below. \begin{verbatim} def seuil(S) : n=0 u=7 while u < S : n=n+1 u=1.05*u+3 return(n) \end{verbatim} Statement 1: the instruction seuil(100) returns the value 18.
Let $(S_n)$ be the sequence defined for every natural integer $n$ by $$S_n = 1 + \frac{1}{5} + \frac{1}{5^2} + \ldots + \frac{1}{5^n}.$$ Statement 2: the sequence $(S_n)$ converges to $\frac{5}{4}$.
Statement 3: in a class composed of 30 students, we can form 870 different pairs of delegates.
We consider the function $f$ defined on $[1 ; +\infty[$ by $f(x) = x(\ln x)^2$. Statement 4: the equation $f(x) = 1$ admits a unique solution in the interval $[1 ; +\infty[$.
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.\\
The five questions of this exercise are independent.
\begin{enumerate}
\item We consider the script written in Python language below.
\begin{verbatim}
def seuil(S) :
n=0
u=7
while u < S :
n=n+1
u=1.05*u+3
return(n)
\end{verbatim}
Statement 1: the instruction seuil(100) returns the value 18.
\item Let $(S_n)$ be the sequence defined for every natural integer $n$ by
$$S_n = 1 + \frac{1}{5} + \frac{1}{5^2} + \ldots + \frac{1}{5^n}.$$
Statement 2: the sequence $(S_n)$ converges to $\frac{5}{4}$.
\item Statement 3: in a class composed of 30 students, we can form 870 different pairs of delegates.
\item We consider the function $f$ defined on $[1 ; +\infty[$ by $f(x) = x(\ln x)^2$.\\
Statement 4: the equation $f(x) = 1$ admits a unique solution in the interval $[1 ; +\infty[$.
\item Statement 5:
$$\int_0^1 x\mathrm{e}^{-x}\,\mathrm{d}x = \frac{\mathrm{e} - 2}{\mathrm{e}}.$$
\end{enumerate}