Is $\sum_{n=3}^\infty \frac{1}{n^2 \log^3 n}$ absolutely convergent?

Question

Is $$\sum_{n=3}^\infty \frac{1}{n^2 \log^3 n}$$ absolutely convergent?

Using comparison test, since $n$ is greater than or equal to 3: $$\frac{1}{n^2 \log^3 n} \leq \frac{1}{n^2}$$

And, we know $$\sum_{n=3}^\infty \frac{1}{n^2}$$ converges by p-test.

Therefore, $$\sum_{n=3}^\infty \frac{1}{n^2 \log^3 n}$$ converges absolutely.

Answer 1

You've answered your own question. Your proof is correct.

Note that in this case "absolutely convergent" is just the same as "convergent" since all the terms are positive.