Well, I like 2048!

Algebra Level 4

Let $a_1,...,a_{2048}$ be non-negative real numbers so that $\sum _{ i=1 }^{ 2048 }{ { a }_{ i } }=1$ . Find the maximum value of $\sum _{ i=1 }^{ 2048 }{ \frac { { a }_{ i } }{ { a }_{ i }^{ 2 }+1 } }$ .

If the answer can be written as $\frac{x}{y}$ , where $x,y$ are positive integers so that $(x,y)=1$ , find $\left| x-y \right|$ .

Bonus: Generalize!

Inspiration.