Is this correct? $c < \log n < n^p < a^n < n! < n^n$ for $a > 1, p > 0$ being used for Direct Comparison test

Question

I mainly ask this because I was thinking about doing Direct Comparison Test for Convergence testing on Series.

I am looking at

$$\sum_{n=2}^{\infty} \frac{(\arctan n)(\ln n)}{n^2}$$

which I am going to compare with

$$\sum_{n=2}^{\infty} \frac{\frac{\pi}{2}(\ln n)}{n^2}$$

can I say that

$$\sum_{n=2}^{\infty} \frac{\frac{\pi}{2}(\ln n)}{n^2} < \sum_{n=2}^{\infty} \frac{n^{0.5}}{n^2} =\sum_{n=2}^{\infty} \frac{1}{n^{1.5}}$$

The final series converges by P series and because that's greater than the original series, the original series also converges.

Did I make any incorrect assumptions?