What is \(\pi\) doing here?

Calculus Level 4

\[ \lim_{n \to \infty} 2^{n+\frac{5}{2}} \sqrt{3-\sqrt{3+3T_{n}}} \]

Define a sequence \( T \) with \( n^\text{th} \) term \( T_{n} \) such that \( T_{0} = 0 \) and \( T_{n} = \sqrt{2+T_{n-1}}\). Then evaluate the limit above.

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