Find the radius of convergence of the following series. Does it converge or diverge? $\infty \sum n = 1 \frac{{(- 1)}^{n} x^{n}}{\sqrt[9]{n}}$ $sum_(n=1)^oo((-1)^nx^n)/root(9)(n)$

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$\infty \sum n = 1 \frac{{(- 1)}^{n} x^{n}}{\sqrt[9]{n}}$ $sum_(n=1)^oo((-1)^nx^n)/root(9)(n)$

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Answer 1 · 2018-04-03T18:33:40

By the ratio test:

$R = lim_(n -> oo) (((-1)^(n + 1)x^(n + 1))/(root(9)(n + 1)))/(((-1)^nx^n)/(root(9)(n))$

$R = lim_(n ->oo) |(-x)| root(9)(n)/root(9)(n + 1)$

The limit in $n$ has a value of $1$ .

Thus

$|x| < 1$

The radius of convergence is $1$ , thus the series converges for $|x| < 1$ .

Hopefully this helps!