Hello can someone help me to calculate this expression? I've tried various ways to find it, but I didn't find the answer.

\[ \lim_{n\to\infty} \left [ \left(\prod_{k=1}^n \dfrac {2k}{2k-1} \right) \int_{-1}^\infty \dfrac{(\cos x)^{2n} }{2^x} \, dx \right ] \]

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