Exercise 3 For each of the following statements, indicate whether it is true or false. Justify each answer. An unjustified answer earns no points.
The sequence $(u_n)$ is defined for every natural integer $n$ by $$u_n = \frac{1 + 5^n}{2 + 3^n}$$ Statement 1: The sequence $(u_n)$ converges to $\frac{5}{3}$.
We consider the sequence $(w_n)$ defined by: $$w_0 = 0 \text{ and, for every natural integer } n,\ w_{n+1} = 3w_n - 2n + 3.$$ Statement 2: For every natural integer $n$, $w_n \geqslant n$.
We consider the function $f$ defined on $]0; +\infty[$ whose representative curve $\mathscr{C}_f$ is given in an orthonormal coordinate system in the figure (Fig. 1). We specify that:
$T$ is the tangent to $\mathscr{C}_f$ at point A with abscissa 8;
The x-axis is the horizontal tangent to $\mathscr{C}_f$ at the point with abscissa 1.
Statement 3: According to the graph, the function $f$ is convex on its domain of definition.
Statement 4: For every real $x > 0$, $\ln(x) - x + 1 \leqslant 0$, where $\ln$ denotes the natural logarithm function.
Exercise 3
For each of the following statements, indicate whether it is true or false. Justify each answer. An unjustified answer earns no points.
\begin{enumerate}
\item The sequence $(u_n)$ is defined for every natural integer $n$ by
$$u_n = \frac{1 + 5^n}{2 + 3^n}$$
Statement 1: The sequence $(u_n)$ converges to $\frac{5}{3}$.
\item We consider the sequence $(w_n)$ defined by:
$$w_0 = 0 \text{ and, for every natural integer } n,\ w_{n+1} = 3w_n - 2n + 3.$$
Statement 2: For every natural integer $n$, $w_n \geqslant n$.
\item We consider the function $f$ defined on $]0; +\infty[$ whose representative curve $\mathscr{C}_f$ is given in an orthonormal coordinate system in the figure (Fig. 1). We specify that:
\begin{itemize}
\item $T$ is the tangent to $\mathscr{C}_f$ at point A with abscissa 8;
\item The x-axis is the horizontal tangent to $\mathscr{C}_f$ at the point with abscissa 1.
\end{itemize}
Statement 3: According to the graph, the function $f$ is convex on its domain of definition.
\item Statement 4: For every real $x > 0$, $\ln(x) - x + 1 \leqslant 0$, where $\ln$ denotes the natural logarithm function.
\end{enumerate}