Counter example for the limit comparison test [closed]
Solution 1:
If $(a_n)$ or $(b_n)$ are eventually positive, then no. If you allow them to have both positive and negative terms infinitely often, then there are counter-examples. For example, consider
$$ a_n = \frac{(-1)^n}{\sqrt{n}} + \frac{1}{2n} \qquad\text{and}\qquad b_n = \frac{(-1)^n}{\sqrt{n}}. $$
Then $a_n/b_n \to 1$ and $\sum_n b_n$ converges by the alternating series test, but $\sum a_n$ diverges.