\textbf{Exercise 3 — 7 points}

Themes: Exponential function and sequence

\textbf{Part A:}

Let $h$ be the function defined on $\mathbb{R}$ by
$$h(x) = \mathrm{e}^x - x$$

\begin{enumerate}
  \item Determine the limits of $h$ at $-\infty$ and $+\infty$.
  \item Study the variations of $h$ and draw up its variation table.
  \item Deduce that: if $a$ and $b$ are two real numbers such that $0 < a < b$ then $h(a) - h(b) < 0$.
\end{enumerate}

\textbf{Part B:}

Let $f$ be the function defined on $\mathbb{R}$ by
$$f(x) = \mathrm{e}^x$$
We denote $\mathscr{C}_f$ its representative curve in a coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.

\begin{enumerate}
  \item Determine an equation of the tangent line $T$ to $\mathscr{C}_f$ at the point with abscissa 0.
\end{enumerate}

In the rest of the exercise we are interested in the gap between $T$ and $\mathscr{C}_f$ in the neighbourhood of 0. This gap is defined as the difference of the ordinates of the points of $T$ and $\mathscr{C}_f$ with the same abscissa. We are interested in points with abscissa $\frac{1}{n}$, with $n$ a non-zero natural number. We then consider the sequence $(u_n)$ defined for all non-zero natural numbers $n$ by:
$$u_n = \exp\left(\frac{1}{n}\right) - \frac{1}{n} - 1$$

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Determine the limit of the sequence $(u_n)$.
  \item a. Prove that, for all non-zero natural numbers $n$,
$$u_{n+1} - u_n = h\left(\frac{1}{n+1}\right) - h\left(\frac{1}{n}\right)$$
where $h$ is the function defined in Part A.\\
b. Deduce the direction of variation of the sequence $(u_n)$.
  \item The table below gives approximate values to $10^{-9}$ of the first terms of the sequence $(u_n)$.
\begin{center}
\begin{tabular}{ | l | c | }
\hline
$n$ & $u_n$ \\
\hline
1 & 0.718281828 \\
\hline
2 & 0.148721271 \\
\hline
3 & 0.062279092 \\
\hline
4 & 0.034025417 \\
\hline
5 & 0.021402758 \\
\hline
6 & 0.014693746 \\
\hline
7 & 0.010707852 \\
\hline
8 & 0.008148453 \\
\hline
9 & 0.006407958 \\
\hline
10 & 0.005170918 \\
\hline
\end{tabular}
\end{center}
Give the smallest value of the natural number $n$ for which the gap between $T$ and $\mathscr{C}_f$ appears to be less than $10^{-2}$.
\end{enumerate}