Consider a sequence $(\varepsilon_n)_{n \in \mathbb{N}}$ of real numbers strictly greater than $-1$, convergent with limit zero. Show that: $$\lim_{n \rightarrow +\infty} \sum_{i=1}^{n-1} \frac{|\varepsilon_i|}{\sqrt{i(n-i)}} = 0.$$
Consider a sequence $(\varepsilon_n)_{n \in \mathbb{N}}$ of real numbers strictly greater than $-1$, convergent with limit zero. Show that:
$$\lim_{n \rightarrow +\infty} \sum_{i=1}^{n-1} \frac{|\varepsilon_i|}{\sqrt{i(n-i)}} = 0.$$