$(n+1)x_{n+2}-nx_{n+1}-x_n=0$ ,prove the sequence converges Sequence $x_n$ for which
$$(n+1)x_{n+2}= nx_{n+1}+x_n$$
for every $n\in\mathbb{N}$. Prove that it converges.
Its not decreasing or increasing, i checked with some random initial values.So, i dont know how to proceed with this. Any help?

The sequence may be rewritten as
$$(n+1) y_{n+1} = -y_n$$
where $y_n = x_{n+1}-x_n$.  This then becomes
$$y_n = \frac{(-1)^n}{n!} y_0$$
or
$$x_{n+1}-x_n = \frac{(-1)^n}{n!} (x_1-x_0)$$
Clearly, as $n \to \infty$, $x_{n+1}-x_n \to 0$ and the sequence converges.
