$ \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.
For what values of x the following series converges?
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calculus
sequences-and-series
limits
1 Answers
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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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0+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. – 2012-11-30