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\[ \lim_{n \to \infty} \left ( \frac {3}{1^2} + \frac {5}{1^2+2^2} + \frac {7}{1^2+2^2+3^2} + \ldots + \frac {2n+1}{1^2+ 2^2 + \ldots + n^2} \right ) = \ ? \]

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