Solving an Inverse-Trigonometric Limit!

Calculus Level 5

\[\large{\lim_{n \to \infty} \left[ \dfrac{1}{n^2} \sum_{1 \leq i < j \leq n} \tan^{-1} \left ( \dfrac{i}{n} \right) \tan^{-1} \left ( \dfrac{j}{n} \right) \right] }\]

If the above limit can be expressed as:

\[\dfrac{\left( \pi^A - \ln(B) \right)^C}{2^D}\]

where \(A,B,C,D\) are positive integers, then find the value of \(A+B+C+D\).

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