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$(-1)^n × 2^{1/n}$

Is it possible to convert this into the form $ar^{n-1}$? I am not so sure on how to convert this. Can someone give me hints or someone guide me in solving this problem. Additionally, if I were to test this series (summation from $n = 1$ to $\infty$ of $(-1)^n × 2^{1/n}$. How would I know if it will converge or diverge?

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salierii
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    You cannot write it in that form. – geetha290krm Dec 06 '23 at 12:28
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    Even ignoring the $-1$ part, the second part -- $2^{\dfrac1n}$ -- isn't a geometric series, because the ratio of adjacent terms isn't a constant. So there's no way to make this look like a geometric series. – John Hughes Dec 06 '23 at 12:29
  • Yup - common ratio is $-\frac{1}{2^{n(n+1)}}$ (not even a common ratio! ratio between consecutive terms I mean) –  Dec 06 '23 at 12:32
  • This is noted. Can I ask, if I were to test this series (summation from n = 1 to infinity of

    (-1)^n * 2^(1/n) . How would I know if it will converge or diverge?

     – salierii Dec 06 '23 at 12:37
  • @salierii The terms approach $\pm1$, so it diverges by the nth term test. – Jam Dec 06 '23 at 12:54

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