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Let's break down both series and analyze their convergence. ### (i) Series: \[ \sum_{n=1}^{\infty} nx^{n-1}, \quad x > 0 \] This is a power series. To analyze its convergence, we can apply the ratio test or use known convergence properties of geometric series. The general term is given by \( a_n = nx^{n-1} \). Applying the ratio test: \[ \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} ____ ___ ______ ___ _______ ____ ______ _________ ______ ____ _________.
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