"Lebesgue" Sum

Calculus Level 4

\[\large\lim_{n\to\infty} \sum_{i=1}^n \frac{i}{n} \left(\sqrt{\frac{i}{n}}-\sqrt{\frac{i-1}{n}}\right)=\frac{a}{b}\]

If the above equation is true for coprime positive integers \(a\) and \(b\), find \(a+b\).

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