Prove $(a_n,b_n >0) \land \sum a_n $ converges $ \land \sum b_n $ diverges$\implies \liminf\limits_{n\rightarrow \infty} \frac{a_n}{b_n}=0$

Question

$(a_n)_{n \in \mathbb{N}},(b_n)_{n \in \mathbb{N}}$ are positive, real sequences!

Since $\sum a_n$ converges, we know $\lim\limits_{n\rightarrow \infty}a_n =0$.

If $\lim\limits_{n\rightarrow \infty}b_n \ne 0$ then $ \liminf\limits_{n\rightarrow \infty} \frac{a_n}{b_n}=0$ is rather easy to show.

So let's assume $b_n$ converges to $0$. Since $\sum a_n$ converges but $\sum b_n$ does not, $a_n$ must somehow "converge faster" to $0$ than $b_n$ does, thus causing $ \liminf\limits_{n\rightarrow \infty} \frac{a_n}{b_n}=0$, but I have a hard time to express that formally.

I'd be very grateful for a push in the right direction.

thanks for helping 🙂

Answer 1

Hint: If $ c = \liminf_{n\to \infty} \frac{a_n}{b_n} > 0$ then $$ a_n \ge \frac c2 b_n \ge 0 $$ for all sufficiently large $n$. What does that tell about the convergence of the series $\sum a_n$ and $\sum b_n$?