We consider $n$ a non-zero natural integer. We consider the function $f_n$ defined on the interval $[0; 1]$ by:
$$f_n(x) = x^n e^{1-x}$$
We admit that the function $f_n$ is differentiable on $[0; 1]$ and we denote $f_n'$ its derivative function.
Part A
In this part we study the case where $n = 1$. We thus study the function $f_1$ defined on $[0; 1]$ by:
$$f_1(x) = x e^{1-x}$$
- Show that $f_1'(x)$ is strictly positive for all real $x$ in $[0; 1[$.
- Deduce the table of variations of the function $f_1$ on the interval $[0; 1]$.
- Deduce that the equation $f_1(x) = 0.1$ admits a unique solution in the interval $[0; 1]$.
Part B
We consider the sequence $(u_n)$ defined for all non-zero natural integers $n$ by
$$u_n = \int_0^1 f_n(x) \, dx \quad \text{that is} \quad u_n = \int_0^1 x^n e^{1-x} \, dx$$
We admit that $u_1 = e - 2$.
- a. Justify that for all $x \in [0; 1]$ and for all non-zero natural integers $n$, $$0 \leq x^{n+1} \leq x^n$$ b. Deduce that for all non-zero natural integers $n$, $$0 \leq u_{n+1} \leq u_n.$$ c. Show that the sequence $(u_n)$ is convergent.
- a. Using integration by parts, prove that for all non-zero natural integers $n$ we have: $$u_{n+1} = (n+1)u_n - 1$$ b. Consider the Python script below defining the function suite(): \begin{verbatim} from math import exp def suite(): u = ... for n in range (1, ...): u = ... return \end{verbatim} Copy and complete the Python script above so that the function suite() returns the value of $\int_0^1 x^8 e^{1-x} \, dx$.
- a. Prove that for all non-zero natural integers $n$ we have: $$u_n \leq \frac{e}{n+1}$$ b. Deduce the limit of the sequence $(u_n)$.