Is the sequence $\frac{1}{\arctan(-n)}\cdot\frac{3n-2}{n^2+n+10}$ increasing or decreasing? Is the following sequence increasing or decreasing $$\frac{1}{\arctan(-n)}\cdot\frac{3n-2}{n^2+n+10}$$?
I managed to come to conclusion that $a_n=\frac{1}{\arctan(-n)}$ is strictly increasing and that for $b_n=\frac{3n-2}{n^2+n+10}, b_1<b_2<b_3>b_4>b_5...$. So $b_n$ is increasing for the first $3$ terms and after that it's decreasing. So what can I conclude about the product of theses $2$ sequences? I know that the product of $2$ increasing sequences doesn't have to be increasing.

Let $(a_n)$ be $\ge 0$ and decreasing, and $(b_n)$ be $\le 0$ and increasing. Then:
$$a_{n+1}b_{n+1} - a_n b_n = a_{n+1}(b_{n+1} - b_n) + (a_{n+1} - a_n)b_n \ge 0$$
So your sequence is increasing.
