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I'm doing a lot of practice problems with sequences, and I've noticed a number of problems ask about the convergence of the sequence raised to a positive power. It seems like in all the examples that I've tried, if $a_n$ converges to $a$, then $a_n^c$ converges to $a^c$, where $c$ is some positive real number. Is this always true?

I want to say yes, since we can define a new sequence $b_n$ as the product of $a_n$ and use the Algebraic Limit Theorem, but I'm wondering if there are any special cases I'm failing to consider.

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    $\begingroup$ There is a problem with definition of $a_n^{c}$ when $a_n <0$. $\endgroup$
    – Kavi Rama Murthy
    May 25, 2020 at 6:39
  • $\begingroup$ I understand! Thank you $\endgroup$
    – kijontrona
    May 25, 2020 at 7:03

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The statement does not hold. For example, take $a_n = -1/n$ and $c=0.5$. $c$ is a positive real number, $a_n$ convergese to $0$, but $a_n^c$ is not defined for all $n$, so the sequence does not converges.

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  • $\begingroup$ Ah, that makes sense. Thank you! $\endgroup$
    – kijontrona
    May 25, 2020 at 7:03

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