We had the following theorem in class:
Let $(a_n)$ and $(b_n)$ be sequences and $b_n>0$ and $\lim_{n\rightarrow\infty}\frac{a_n}{b_n}=L$ with $L\in\mathbb R\backslash\{0\}$. Then $\sum a_n$ converges if and only if $\sum b_n$ converges.
So by recapitulating the lecture I've tried to prove it but I didn't get it. It's obvious that you have to use the comparison test, but how? Can anybody help? Thanks a lot!
Best Answer
Hint: Suppose that $L\gt 0$ (a similar argument will deal with $L\lt 0$).
Because $\lim_{n\to\infty}\frac{a_n}{b_n}=L$, there is an $N$ such that if $n\gt N$ then $$\frac{L}{2}\lt \frac{a_n}{b_n}\lt \frac{3L}{2}.$$ The above inequality follows from the definition of limit by taking $\epsilon=\frac{L}{2}$.
Now, as you expected, Comparison does it.