We are given the given sequence:
$$ x_1 = 1$$ and for all positive integers $n \ge 2 $ :
$$ n (x_n)^2 - x_{n-1} - n = 0 $$ where $ x_n \ge 0$ for all $ n \ge 1 $
We are supposed to find the limit as $ n \to \infty $ of $x_n$ and hence show that the series sum diverges as we sum up to infinity.
I noticed that as the terms go on and on, the value of $x_n$ gets closer to $1$.
Furthermore, I was able to manipulate the relation to arrive at:
$$ x_n = \sqrt{1 + \frac{1}{n} \ x_{n-1}} $$
So I need to show that as $ n \to \infty $, the term $ \frac{x_{n-1}}{n} $ tends to $0$.
However, I was stuck at this part and I cannot seem to proceed further. Any help on this problem and/or alternative approaches will be greatly appreciated!
By definition we have $x_n\ge0$. We also have the somewhat trivial upper bound given from
$$x_n\le\sqrt{1+x_{n-1}}$$
as well as the inductively applied inequalities
$$x\le\varphi\implies\sqrt{1+x}\le\sqrt{1+\varphi}=\varphi$$
where $\varphi=\dfrac{1+\sqrt5}2$ is the golden ratio. Since it is bounded we have the results from the squeeze theorem
$$0\le\frac{x_{n-1}}n\le\frac\varphi n\to0$$
implying that
$$x_n\to\sqrt{1+0}=1$$
as $n\to\infty$.