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

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?)

## marked as duplicate by Chinnapparaj R, Cm7F7Bb, Community♦Dec 5 '18 at 13:10

• This is true for series with a posiitve (or negative) general term. It is the criterion by equivalence of functions. – Bernard Dec 5 '18 at 13:00
• Thank you very much everyone for your time, turns out it was a duplicate.. – Boaz Yakubov Dec 5 '18 at 13:11

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