4. We have

$$
\begin{aligned}
& a_{n}=\prod_{k=2}^{n} \frac{k^{2}}{k-1}=\frac{2 \cdot 2}{1} \cdot \frac{3 \cdot 3}{2} \cdot \ldots \ldots . \cdot \frac{n \cdot n}{n-1}=\frac{n!\cdot n!}{(n-1)!}=n!\cdot n=n!\cdot[(n+1)-1]=(n+1)!-n! \\
& b_{n}=\sum_{k=2}^{n}[(k+1)!-k!]=(n+1)!-2, c_{n}=\frac{(n+1)!-2}{(n+1)!}=1-\frac{2}{(n+1)!} \\
& d_{n}=\left(1+\frac{-2}{(n+1)!}\right)^{(n+1)!}
\end{aligned}
$$

$$
\begin{gathered}
\text { We have } \lim _{n \rightarrow \infty} c_{n}=\lim _{n \rightarrow \infty}\left(1+\frac{-2}{(n+1)!}\right)=1+0=1 \text { and } \lim _{n \rightarrow \infty} d_{n}=\lim _{n \rightarrow \infty}\left(1+\frac{-2}{(n+1)!}\right)^{(n+1)!}= \\
=\lim _{n \rightarrow \infty}\left[\left(1+\frac{-2}{(n+1)!}\right)^{\frac{(n+1)!}{-2}}\right]^{\frac{-2}{(n+1)!}(n+1)!}=e^{-2}=\frac{1}{e^{2}} \quad \text {.................................................................................... }
\end{gathered}
$$

## MATHEMATICS OLYMPIAD  Local stage, 16.02.2013  11th Grade

## Problems:
4. Consider the sequences of real numbers $\left(x_{n}\right)_{n \geq 2},\left(a_{n}\right)_{n \geq 2},\left(b_{n}\right)_{n \geq 2},\left(c_{n}\right)_{n \geq 2},\left(d_{n}\right)_{n \geq 2},$ with the general terms defined as follows:

$$
\begin{gathered}
x_{n}=\left(n+\frac{n}{n-1}\right) \\
a_{n}=x_{2} x_{3} \ldots x_{n}, \quad b_{n}=a_{2}+a_{3}+\cdots+a_{n}, c_{n}=\frac{b_{n}}{(n+1)!}, d_{n}=\left(c_{n}\right)^{(n+1)!}
\end{gathered}
$$

Calculate the limits of the sequences $\left(c_{n}\right)_{n \geq 2}$ and $\left(d_{n}\right)_{n \geq 2}$.

Mihai Bodan, Cosmești, Teleorman