Let \( m \) and \( n \) be two positive integers greater than 1. If...

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Let \( m \) and \( n \) be two positive integers greater than 1. If \( \lim _{\alpha \rightarrow 0}\left(\frac{e^{\cos \left(\alpha^{n}\right)}-e}{\alpha^{m}}\right)=-\left(\frac{e}{2}\right) \), then \( \frac{m}{n} \) is
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