Word Count : 267

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आइए हम दोनों श्रेणियों के अभिसरण की जांच करें: ### (i) \( \sum_{n=1}^{\infty} n x^{n-1}, \quad x > 0 \) इस श्रेणी को एक भिन्न रूप में लिखा जा सकता है: \[ S(x) = \sum_{n=1}^{\infty} n x^{n-1} = \frac{1}{(1-x)^2} \quad \text{जब} \quad |x| < 1 \] यह श्रेणी एक अपरिपक्व ज्यामितीय श्रेणी की ______ ______ ____ ___ ____ _______ ______.
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