Marinada '22

[lang=hr] Lovri je sada već teško pomaknuti se nakon 2 boce dedinog soka od šljiva, polako mu se sklapaju oči i kreće spavati. Budući da Lovro u sebi ima matematičarske gene on sanja sljedeći zadatak: Neka je $n$ prirodan broj. Neka je $x_0 = 0$ i neka je $x_i > 0 $ za svaki $i \in {1,2,3,...,n}$. Ako je $x_1+x_2+...+x_n = 1$ odredi najmanji $M$ tako da vrijedi $$\sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+x_1+\ldots +x_{i-1}}\sqrt{x_i+\ldots+x_n}} < M$$. Odgovor zaokružite na 6 decimala. [/lang] [lang=en] It's already hard for Lovro to move after 2 bottles of grandfather's plum juice, his eyes slowly close and he goes to sleep. Since Lovro has mathematical genes in him, he dreams of the following task: Let $n$ be a natural number. Let $x_0 = 0$ and let $x_i > 0 $ for every $i \in {1,2,3,...,n}$. If $x_1+x_2+...+x_n = 1$, determine the smallest $M$ such that the following is true; $$\sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+x_1+\ldots +x_{i-1}}\sqrt{x_i+\ldots+x_n}} < M$$. Round the answer to 6 decimal places. [/lang]