find a the following series this is just somthing i thought about (i dont know if there is an answer)

Let $ \sum_{n=1}^{\infty}a_n \to L_1$
($a_n $ is a positive sequence)
find a sequence $b_n$ such that:
$\lim_{n\to\infty} \dfrac{b_n}{a_n} = \infty$
and $\sum_{n=0}^{\infty}b_n = L_2$ (the series converges)

if there is no solution can this be proven?
if we add the fact that $a_n$ is monotone decending would that help?
thanks alot!

NOTICE
This is an answer to the original question. While I was writing the answer the OP changed the condition $a_n/b_n\to\infty$ to $b_n/a_n\to\infty$.
Take $b_n=a_n^2$. Since $\sum a_n$ converges, $a_n\to0$, and $a_n/b_n=1/a_n\to\infty$. Also, $a_n$ is bounded. Let $A$ be an upper bound. Then $0\le b_b\le A\,a_n$, so that $\sum b_n$ converges.
