I've seen limit comparison test to be used many times, but there exist a version of this test named '' one sided limit comparison test" which states:
If $a_{n},b_{n}\ge0$ for all $n$ and $\limsup _{n\to\infty}\frac{a_{n}}{b_{n}}=c$ where $c$ is a finite nonnegative real number, then if $\sum_{n}^{ }b_{n}$ is convergent then so is $\sum_{n}^{ }a_{n}$.
The converse of this statement is as follows :
If $a_{n},b_{n}\ge0$ for all $n$ and if $\sum_{n}^{ }a_{n}$ diverges but $\sum_{n}^{ }b_{n}$ converges, then necessarily $\limsup _{n\to\infty}\frac{a_{n}}{b_{n}}=\infty$ or equivalently $\liminf _{n\to\infty}\frac{b_{n}}{a_{n}}=0$
I've never seen any proof about these two statements, can someone give me a link or a proof?
For the first question: $\frac {a_n} {b_n}$ is bounded, say by $M$ and this implies that $\sum a_n \leq M \sum b_n <\infty$.
For the second question: Suppose $\sum a_n$ diverges and $\sum b_n$ converges. Let us prove by contradiction that $\lim \sup \frac {a_n} {b_n} =\infty$. If $\lim \sup \frac {a_n} {b_n} <\infty$ then there exist $M$ and $n_0$ such that $\frac {a_n} {b_n} <M$ for all $n >n_0$. Since $\sum b_n <\infty$ and $a_n<M b_n$ for all $n >n_0$ it follows that $\sum a_n <\infty$ which is a contradiction. Hence $\lim \sup \frac {a_n} {b_n} $ must be $\infty$.