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Let \( \mathfrak P_n=\dfrac 1n+\dfrac{1}{3n^3}+\dfrac{1}{5n^5}+\dfrac{1}{7n^7} + \cdots\).

If the value of \(\displaystyle\sum_{n=2}^{99}\ 2\mathfrak{P}_n\) is equal to \( \ln\alpha\), find \(\alpha\).

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