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 \(\left| x-y \right| \).