Crazy Convergence

Calculus Level 5

$\large \sum_{k=1}^n \frac 1 {k^2 x - 1} = \frac 1 2$

Consider the equation above, for every natural number $$n$$, this equation admits a unique solution $$x_n > 1$$ (One can prove this).

Suppose the sequence $$x_n$$ converges to $$l$$, find $$l^{2}$$.

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