# 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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