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?)

This is true for series with a posiitve (or negative) general term. It is the criterion by equivalence of functions. — Bernard, Dec 05 '18 at 13:00
Thank you very much everyone for your time, turns out it was a duplicate.. — Boaz Yakubov, Dec 05 '18 at 13:11

TheD0ubleT · Answer 1 · 2018-12-05T16:11:17.113

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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

edited Dec 05 '18 at 16:11

answered Dec 05 '18 at 12:52

TheD0ubleT

376

1

@A.Pongrácz For series of positive numbers this is a complete proof. In the example you give, both series diverge. – MaoWao Dec 05 '18 at 13:09

mathcounterexamples.net · Answer 2 · 2018-12-05T13:34:09.807

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

edited Dec 05 '18 at 13:34

answered Dec 05 '18 at 12:59

mathcounterexamples.net

70,018

You're right I fixed it. – mathcounterexamples.net Dec 05 '18 at 13:37

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

2 Answers2