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To test the series for absolute and conditional convergence, we are given the series: \[ S(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{\sin(nx)}{n\sqrt{n}}. \] ### Step 1: Test for Absolute Convergence For absolute convergence, we need to consider the series formed by the absolute values of the terms: \[ S_{\text{abs}}(x) = \sum_{n=1}^{\infty} \left| (-1)^{n-1} \frac{\sin(nx)}{n\sqrt{n}} \right| = \sum_{n=1}^{\infty} \frac{|\sin(nx)|}{n\sqrt{n}}. \] Since \( |\sin(nx)| \leq 1 \) for all \( n \), the series simplifies to: \[ S_{\text{abs}}(x) \leq ________ ____ __________ ____ __________ _______ ____ ______ ____ ______ ________.
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