3. (7p) Consider the sequence $\left(x_{n}\right)_{n \in N}$ with $x_{0} \geq 0, a \in(0,1)$ and $x_{n+1}=a x_{n} e^{-x_{n}}, n \in N$. Calculate

$$
\lim _{n \rightarrow \infty} x_{n} \ln \left(1+n^{3}\right)
$$

GM10/2015
3. We denote $\lim _{n \rightarrow \infty} x_{n} \ln \left(1+n^{3}\right)=l$ (if the limit exists)

If $x_{0}=0$, then $x_{n}=0, \forall n \in N$, so $l=0$

Let $x_{0}>0$; then (by induction) $x_{n}>0, \forall n \in N$

We have $\frac{x_{n+1}}{x_{n}}=a \cdot e^{-x_{n}}<a, \forall n \in N$, hence $x_{n} \leq a^{n} x_{0}, \forall n \in N$

It follows that $0 \leq x_{n} \ln \left(1+n^{3}\right) \leq a^{n} x_{0} \ln \left(1+n^{3}\right), \forall n \in N$ (1)

Using $\lim _{x \rightarrow \infty} \frac{\ln x}{x}=0$ and $\lim _{n \rightarrow \infty} a^{n} P(n)=0, P$ a polynomial function, we have

$$
\lim _{n \rightarrow \infty} a^{n} x_{0} \ln \left(1+n^{3}\right)=\lim _{n \rightarrow \infty}\left(x_{0} \cdot \frac{\ln \left(1+n^{3}\right)}{1+n^{3}} \cdot a^{n}\left(1+n^{3}\right)\right)=0
$$

(1p)

From (1) it follows that $l=0$