Prove/Disprove $\sum a_n$ and $\sum b_n$ converge together iff $\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}/b_n } = 1$

Question

Prove/Disprove that if $\sum a_n$ and $\sum b_n$ are some series and $\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}/b_n } = 1$ then the series converge together or not converge together.

This doesn't seem to be correct to me so maybe there is some counter example. i know this is true if both series are strictly nonnegative (from the first series comparison test) - how does it change if both are negative (or alternate?)

Answer 1

If $\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}/b_n } = 1$ , then $$(\exists N\in \Bbb N, \forall n > N):\; \frac{1}{2}a_n \le b_n\le \frac{3}{2}a_n$$

Thus if $\sum a_n$ converges then so does $\sum b_n$ and vice versa

Update As stated in the comment this is infact only true if $a_n$ and $b_n$ keep a constant sign (positive or negative) after a certain n. Otherwise it's false: cf. the counter example given below

Answer 2

A counterexample is: $$a_n = \frac{(-1)^n}{\sqrt{n}} \text{ and } b_n = \frac{(-1)^n}{\sqrt{n} + (-1)^n}.$$