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Let $\sum_2^\infty a_nx^n$ be a power series. Find the radius of convergence when $\lim \limits_{n \to \infty} \frac {a_n}{3^n}$ = 1.

There are a few more questions in this manner, but I'd like to understand how to tackle a simple one with some help before I go onto the harder ones. I'm guessing the root test will be used in this.

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You have

$$\lim_{n\to \infty} \sqrt[n]{a_n} = \lim_{n\to \infty} {3}\frac{1}{3}\sqrt[n]{a_n}= \lim_{n\to \infty} {3}\sqrt[n]{\frac{a_n}{3^n}} = 3$$

So, with root test for power series, $R=\frac{1}{3}$

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  • $\begingroup$ Could you help me with a follow-up? Now $\lim \limits_{n \to \infty} \frac {a_n}{n^3}$ = 1. Find R $\endgroup$ – Ronique Hossain Mar 2 '15 at 21:42

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