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To analyze the series and test its convergence, we need to examine it numerically. The series in question is: \[ \sum_{n=1}^{\infty} (-1)^{n-1} \frac{\sin(nx)}{n\sqrt{n}} \] ### Absolute Convergence: For absolute convergence, we check if the series of absolute values converges. That is, we examine: \[ \sum_{n=1}^{\infty} \left| \frac{\sin(nx)}{n\sqrt{n}} \right| \] This is equivalent to: \[ \sum_{n=1}^{\infty} _______ ____ ______ ____ _____ _________ _______ ______ ______.
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