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$$ \sum_{n=1}^{\infty} \frac{(-1)^n \, n! \, x^n}{10^n} $$ For what values ​​of x converges?. I tested the criteria of reason and root, but both give me $\infty$, I do not know how to interpret it.

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up vote 3 down vote accepted

At $x=0$ it certainly converges!

For $x\ne 0$, show that $$\frac{n!|x|^n}{10^n}$$ does not approach $0$. (In fact, it blows up.). Then recall that if the terms $a_n$ do not approach $0$, then the series $\sum_1^\infty a_n$ does not converge.

This will not be difficult. Rewrite the expression as $$\frac{n!}{(10/|x|)^n}.\tag{$1$}$$ As soon as $n \gt 10/|x|$, the expression $(1)$ starts to increase. This can be checked by looking at the ratio of successive terms.

You have in essence done this already. The fact that the Ratio Test gives "$\infty$" means that the series diverges.

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+1, but I think it may be helpful to also include the general form for a power series and how it can converge at $x = c$ only here, as your first line eludes. I feel this makes it a little easier to see as a reader, that's all. – Joe Nov 30 '12 at 2:18

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