Is the series conditionally convergent, absolutely convergent or divergent $\sum(-1)^n\frac{\ln^3 n}n$.

Question

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?

Answer 1

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

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