Let the string $(x_n)_{n\geq 0}$ such that : $$\frac{1}{1}, \frac{1}{2}, \frac{1}{2}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}\cdots$$
Find $$\lim_{n\to\infty} x_n\cdot\sqrt{n}.$$
Hello. I just found that problem and i have no ideea on solving it. Actually we can see that $x_{\frac{n(n+1)}{2}} = \frac{1}{n}$, but i don't really got nothing. Somebody can help me? Thank you!
$\textbf{Hint :}$
We have : $$ \left(\forall n\in\mathbb{N}\right)\left(\forall k\in\left[\!\left[0,n\right]\!\right]\right),\ x_{\frac{n\left(n+1\right)}{2}+k}=\frac{1}{n+1} $$
A closed form for $ \left(x_{n}\right)_{n\geq 0} $ can be given as follows : $$ \left(\forall n\in\mathbb{N}\right),\ x_{n}=\frac{1}{\left\lfloor\frac{\sqrt{1+8n}-1}{2}\right\rfloor +1} $$