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How to find the radius of convergence and interval of convergence of the following power series?

$$\sum{x^{n!}}$$

and

$$ \sum{\frac{1}{n^{\sqrt{n}}} x^n} $$

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    A [related question](http://math.stackexchange.com/q/228852/26872).2012-12-08

2 Answers 2

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For the first one just use the ratio test:

$$\frac{x^{(n+1)!}}{x^{n!}}=x^{(n+1)!-n!}=x^{n!((n+1)-1)}=x^{n\cdot n!}\;.$$

Now what’s $$\lim_{n\to\infty}|x|^{n\cdot n!}\;?$$

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    @Babak: Shh! ;-) Yes, it is.2012-12-08
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Hint: Use the root test for the second series and note that

$$ \lim_{n\to \infty} \left(\frac{1}{n^{\sqrt{n}}}\right)^{1/n}=1. $$

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    Hint:I think the following link will be helpful to answer the first question...http://math.stackexchange.com/questions/252833/finding-the-radius-of-convergence-of-sum-0-inftyzn/252839#2528392012-12-08
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    @learner: I did not answered the first!!2012-12-08
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    sorry,sir.I apologize..2012-12-08