For what $x \in \mathbb{R}$ does the series $\sum_{n=0}^{\infty} n!x^{n}$ converge?
How would you prove this? Does the ratio test apply?
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2"Does the ratio test apply?" : Try and you'll see !!! But Ratio test always works... like all other test. But it's not enough to determine all $x$ (you need to check what happen on the bound) – Surb Sep 24 '16 at 22:29
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@Surb not in this case. – quid Sep 24 '16 at 22:30
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@quid: yes it does. – Surb Sep 24 '16 at 22:31
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@Surb I meant that when the radius of convergence given by the ratio test is $0$ then there is no need to check anything "on the boundary." It's not quite clear what you assumed I was saying. As yeah the ratio test applies, as I said in my answer. – quid Sep 24 '16 at 22:34
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........ 0 ........ – Mark Viola Sep 25 '16 at 00:04
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Linking to a duplicate. – metamorphy Nov 09 '20 at 03:40
1 Answers
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Yes, the ratio test applies and yields that it converges for $x=0$ only.
Let me know if you need more details.
quid
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