Exercise 1 -- Common to all candidates
Part A
Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = x\mathrm{e}^{1-x}$$
- Verify that for all real $x$, $f(x) = \mathrm{e} \times \frac{x}{\mathrm{e}^{x}}$.
- Determine the limit of the function $f$ at $-\infty$.
- Determine the limit of the function $f$ at $+\infty$. Interpret this limit graphically.
- Determine the derivative of the function $f$.
- Study the variations of the function $f$ on $\mathbb{R}$ then draw up the variation table.
Part B
For every non-zero natural number $n$, we consider the functions $g_n$ and $h_n$ defined on $\mathbb{R}$ by: $$g_n(x) = 1 + x + x^2 + \cdots + x^n \quad \text{and} \quad h_n(x) = 1 + 2x + \cdots + nx^{n-1}.$$
- Verify that, for all real $x$: $(1-x)g_n(x) = 1 - x^{n+1}$.
We then obtain, for all real $x \neq 1$: $g_n(x) = \frac{1 - x^{n+1}}{1-x}$.
2. Compare the functions $h_n$ and $g_n'$, $g_n'$ being the derivative of the function $g_n$.
Deduce that, for all real $x \neq 1$: $h_n(x) = \frac{nx^{n+1} - (n+1)x^n + 1}{(1-x)^2}$.
3. Let $S_n = f(1) + f(2) + \ldots + f(n)$, $f$ being the function defined in Part A.
Using the results from Part B, determine an expression for $S_n$ then its limit as $n$ tends to $+\infty$.