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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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