Three Sums

Algebra Level 5

For a positive integer \(k\), let \(x_k\) be the largest positive real solution to the equation \[\frac{2}{x}+\frac{1}{\sqrt{k(k+1)x^2-1}}=1.\] Let \[S_n=\sum_{k=1}^n\frac{1}{\sqrt{2+2x_k}}, \quad T_n=\sum_{k=1}^n\frac{1}{1+\sqrt{2}+2S_k}\]
and \[U_n=\sum_{k=1}^n\left(T_k^2+\frac{k^2}{T_k^2}\right).\] Find the least positive integer \(n\) such that \(U_n>201600\).

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