Word Count : 377

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To show that the series \[ \sum_{n=1}^{\infty}\frac{x}{1+n^2x^2} \] is uniformly convergent on the interval \([a, 1]\) for any \( a > 0 \), we need to analyze its convergence and uniformity on the given interval. ### Step 1: Study the behavior of the general term The general term in the series is: \[ f_n(x) = \frac{x}{1 + n^2 x^2} \] For any fixed \(x \in [a, 1]\), we know that the terms in the series ___ ________ _____ _____ ___ ____.
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