We consider the sequence $(u_n)$ defined by: $$\left\{\begin{aligned} u_1 &= \frac{1}{\mathrm{e}} \\ u_{n+1} &= \frac{1}{\mathrm{e}}\left(1 + \frac{1}{n}\right)u_n \text{ for all integer } n \geqslant 1. \end{aligned}\right.$$
Calculate the exact values of $u_2$ and $u_3$. Details of calculations should be shown.
We consider a function written in Python language which, for a given natural integer $n$, displays the term $u_n$. Complete lines $L_2$ and $L_4$ of this program. \begin{verbatim} L1 def u(n): L2 ................. L3 for i in range(1, n): L4 u=................. L5 return u \end{verbatim}
We admit that all terms of the sequence $(u_n)$ are strictly positive. a. Show that for all non-zero natural integer $n$, we have: $1 + \dfrac{1}{n} \leqslant \mathrm{e}$. b. Deduce that the sequence $(u_n)$ is decreasing. c. Is the sequence $(u_n)$ convergent? Justify your answer.
a. Show by induction that for all non-zero natural integer, we have: $u_n = \dfrac{n}{\mathrm{e}^n}$. b. Deduce, if it exists, the limit of the sequence $(u_n)$.
We consider the sequence $(u_n)$ defined by:
$$\left\{\begin{aligned} u_1 &= \frac{1}{\mathrm{e}} \\ u_{n+1} &= \frac{1}{\mathrm{e}}\left(1 + \frac{1}{n}\right)u_n \text{ for all integer } n \geqslant 1. \end{aligned}\right.$$
\begin{enumerate}
\item Calculate the exact values of $u_2$ and $u_3$. Details of calculations should be shown.
\item We consider a function written in Python language which, for a given natural integer $n$, displays the term $u_n$. Complete lines $L_2$ and $L_4$ of this program.
\begin{verbatim}
L1 def u(n):
L2 .................
L3 for i in range(1, n):
L4 u=.................
L5 return u
\end{verbatim}
\item We admit that all terms of the sequence $(u_n)$ are strictly positive.\\
a. Show that for all non-zero natural integer $n$, we have: $1 + \dfrac{1}{n} \leqslant \mathrm{e}$.\\
b. Deduce that the sequence $(u_n)$ is decreasing.\\
c. Is the sequence $(u_n)$ convergent? Justify your answer.
\item a. Show by induction that for all non-zero natural integer, we have: $u_n = \dfrac{n}{\mathrm{e}^n}$.\\
b. Deduce, if it exists, the limit of the sequence $(u_n)$.
\end{enumerate}