Comparison test $\sum_{n=1}^{\infty} \frac{1}{2n+7}$
My textbook did the limit comparison test to do this but cant you just do this using direct comparison test?
What I did
For $n \geq 1, \forall n \in \mathbb N$, $a_n = \frac{1}{2n+7} \leq \frac{1}{n} = b_n$
Since $$0 < a_n \leq b_n, \forall n \in \mathbb N$$
Direct comparison test applies
Consider $\sum_{n=1}^{\infty} \frac{1}{n}$. Since this is a p-series with magnitude one we know that $\sum b_n$ diverges. Therefore by the direct comparison test $\sum a_n$ diverges as well.
Usually my textbook does the easiest way so I don't know.