$$\sum_{n=1}^\infty \frac{2n-1}{4n^2+1}\tag{1}$$
i would like to find out if this series convergent or not so i use Limit comparison test and choose $a_n$ and $b_n=\frac{1}{n}$ why do i need to choose $b_n$ as $\frac{1}{n}$ because at the end $\frac{1}{n}$ is divergent and whole $a_n$ is divergent.???? why can't i choose $b_n$ as $\frac{1}{n^2}$?