Roots and Riemann

Calculus Level 3

\[\large \lim_{n \to\ \infty} \left(\frac{1}{\sqrt{n}\sqrt{n + 1}} + \frac{1}{\sqrt{n}\sqrt{n + 2}} + \cdots + \frac{1}{\sqrt{n}\sqrt{n + n}} \right)\]

The value of the limit above can be expressed as \(\displaystyle a\sqrt{b} - \frac cd\), where \(a\), \(b\), \(c\), and \(d\) are positive integers, \(\gcd(c, d) = 1\), and \(b\) is square-free.

Find \(a^4 + b^3 + c^2 + d\).

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