Word Count : 165

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(i) Consider the series $\sum_{n=1}^{\infty} n x^{n-1}, x>0$. We can rewrite it as $\sum_{n=1}^{\infty} n x^{n-1} = \sum_{n=0}^{\infty} (n+1) x^n$. To test for convergence, we apply the ratio test: $$ a_n = (n+1)x^n, \quad a_{n+1} = (n+2)x^{n+1} $$ $$ \frac{a_{n+1}}{a_n} = _________ ___ ____ ______ _________ ________ _________.
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