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Find the interval $I$ of convergence of the series

$$\sum_{n=1}^{\infty} 9 (-1)^n n x^n.$$

(Enter your answer using interval notation.)

I'm stuck with this problem and could use a ton of help.

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2 Answers

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$$\left|\frac{a_{n+1}}{a_n}\right|=\frac{n+1}n|x|\xrightarrow[n\to\infty]{}|x|$$

So it has to be...

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I do not know how much this could help; so forgive me if I am off-topic.

$$S=\sum_{n=1}^{\infty} 9 (-1)^n n x^n=9x\sum_{n=1}^{\infty} (-1)^n n x^{n-1}=9x\frac{d}{dx}\Big(\sum_{n=1}^{\infty} (-1)^n x^{n}\Big)$$ $$\sum_{n=1}^{\infty} (-1)^n x^{n}=-1+\sum_{n=0}^{\infty} (-1)^n x^{n}=-1+\frac{1}{x+1}=-\frac{x}{x+1}$$ Derive and get $$S=-\frac{9 x}{(x+1)^2}$$ from which the problem becomes quite clear.

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