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Determine if the series

$$\sum_{n=1}^\infty(-1)^n\frac{\ln^3 n}n$$

is conditionally convergent, absolutely convergent or divergent.

By comparison test I got $a_n > b_n$ therefore divergent with $a_n=\frac{\ln^3 n}n$ and $b_n=\frac 1n$.

Then with the alternating test I concluded it converges.

Therefore the original series is conditionally convergent.

Is my though process correct here or is there correction needed?

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  • $\begingroup$ What are $a_n$ and $b_n$? $\endgroup$ Commented Oct 3, 2020 at 14:08
  • $\begingroup$ a_n: (ln (n))^3 / n. b_n: 1/n $\endgroup$ Commented Oct 3, 2020 at 14:13
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    $\begingroup$ Yes. You are correct. $\endgroup$
    – Mark Viola
    Commented Oct 3, 2020 at 14:47

1 Answer 1

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Yes your though process is correct since eventually $\log^3 n \ge 1$ for the absolute series we have

$$\frac{\ln^3 n}n \ge \frac1n$$

which diverges by comparison test, while the alternating series converges by alternating series test since eventually

$$f(x)=\frac{\ln^3 x}x \implies f'(x)=-\frac{(\log x -3)\log^2 x}{x^2}<0$$

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