Word Count : 251

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To show that the series \[ \sum_{n=1}^{\infty} \frac{x}{1 + n^2 x^2} \] is uniformly convergent in \([ \alpha, 1 ]\) for any \(\alpha > 0\), we apply the Weierstrass M-test. ### Step 1: Find a Dominant Sequence Define the general term of the series: \[ f_n(x) = \frac{x}{1 + n^2 x^2} \] for \(x \in [\alpha, 1]\), where \(\alpha > 0\). To __________ ___ _____ _____ _______ _________ _________ __________ __________ _____ ____.
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