Use comparison test to determine divergence/convergence

Question

$$A_n=\sum_{n=1}^{\infty}\frac{4}{7n+4\sqrt{n}}$$

How to choose $$B_n$$ so that this problem can be solved by comparison test??

Answer 1

$$A_n=\sum_{n=1}^{\infty}\frac{4}{7n+4\sqrt{n}}\gt \sum_{n=1}^{\infty}\frac{4}{7n+4n}=\sum_{n=1}^{\infty}\frac{4}{11n}$$.

By Comparison Test, $$\sum_{n=1}^{\infty}\frac{4}{11n}$$ diverges. Hence so does $$A_n$$.

Answer 2

Here is an approach. Notice this

$$ \frac{4}{7n+4\sqrt{n}}\sim_{\infty} \frac{4}{7n}=b_n. $$

Then use the result:

Suppose $\sum_{n} a_n$ and $\sum_n b_n $ are series with positive terms, then

if $\lim_{n\to \infty} \frac{a_n}{b_n}=c>0$, then either both series converge or diverge.