It looks messy, but also familiar

Algebra Level 3

\[\large \lim_{n\to\infty} \left [ \log_n \left(1 + \sum_{m=1}^{n^2-1} \dfrac1{\sqrt{m+1} + \sqrt m} \right) \right ] = \, ? \]

Let \(n\) be a positive integer greater than 1. Evaluate the limit above.

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