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Does the infinite series $\sum_{n=2} \frac{(-1)^n}{\sqrt[n]{ln(n)}}$ converge absolutely / converge / diverge?

I can show the the positive and negative element series diverge, so I know the series does not converge absolutely, but I don't know how to tell if it converges.

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Note that

$$\frac{1}{\sqrt[n]{\ln n}}\ge \frac{1}{\sqrt[n]{n}}\to 1$$

then the series doesn't converge.

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  • $\begingroup$ I actually thought of this but for some reason just forgot to apply that my original series is larger then this series (brain fart). Thanks! $\endgroup$ – Jason May 17 '18 at 17:13
  • $\begingroup$ @Jason Yes it is always the first check to do! $\endgroup$ – gimusi May 17 '18 at 17:14
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We have $$ \frac{1}{\sqrt[n]{\ln n}} = e^{-\frac{1}{n}\ln\ln n} \xrightarrow[n\to\infty]{} e^0 = 1 $$ so the general term does not converge to $0$: the series diverges by the term test.

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  • $\begingroup$ This leads the OP so faster. $\endgroup$ – mrs May 17 '18 at 17:15

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