True/False: $\mathop {\lim }\limits_{n \to \infty } {{{a_n}} \over {{b_n}}} = 1$ implies $\sum {{a_n},\sum {{b_n}} } $ converge or diverge together.

Question

$$\mathop {\lim }\limits_{n \to \infty } {{{a_n}} \over {{b_n}}} = 1$$ Prove the statement implies $\sum {{a_n},\sum {{b_n}} } $ converge or diverge together.
My guess the statement is true.

if $\sum{{a_n}}$ diverges, then $\mathop {\lim }\limits_{n \to \infty } {a_n} \ne 0$

So, $$\eqalign{ & \mathop {\lim }\limits_{n \to \infty } {a_n} = L \ne 0 \cr & {{\mathop {\lim }\limits_{n \to \infty } {a_n}} \over {\mathop {\lim }\limits_{n \to \infty } {b_n}}} = 1 \Rightarrow {L \over {\mathop {\lim }\limits_{n \to \infty } {b_n}}} = 1 \Rightarrow L = \mathop {\lim }\limits_{n \to \infty } {b_n} \ne 0 \cr} $$

therefore, $\sum {b_n}$ also diverges.

What I was not managed to do is proving that the two series converges together.
Or maybe the statement is not always true?

$\sum 1/n$ diverges and yet $\lim_{n\to\infty} 1/n=0$, so your proof does not work. — Andrés E. Caicedo, Dec 05 '13 at 21:37
Yes, the statement "If $\sum a_n$ diverges, then $\lim a_n\neq 0$" is false. — Thomas Andrews, Dec 05 '13 at 21:38
Lim $a_n/b_n=1$ then $\exists N$ s.t. $a_n/b_n>1/2$ for $n>N$ this implies $b_n<2a_n$ if $\sum a_n$ converges, so does $\sum b_n$ and vice versa. — derivative, Dec 05 '13 at 21:46
@AndresCaicedo Thanks. Here's an counter-example http://math.stackexchange.com/a/30549/99325 — derivative, Dec 05 '13 at 23:16
@Daniel Gagnon : Are your $a_n$'s and $b_n$'s positive? This might be important. — Stefan Smith, Dec 06 '13 at 02:39
@Daniel Gagnon : your assertion that "if $\sum a_n$ diverges, then $\lim_{n\to\infty} \neq 0$" is incorrect, and there is a familiar counterexample. — Stefan Smith, Dec 06 '13 at 02:44

score 13 · Accepted Answer · answered Dec 06 '13 at 16:22

13

Surprisingly, this statement is false. For a simple counter-example, consider $$ a_n = \frac{(-1)^n}{\sqrt{n}},\quad\text{and}\quad b_n = \frac{(-1)^n}{\sqrt{n}} + \frac{1}{n} $$ The condition $a_n \sim b_n$ holds but $\sum a_n$ is convergent whereas $\sum b_n$ is divergent.

answered Dec 06 '13 at 16:22

Siméon

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(Thus, the hypothesis that $a_n,b_n\geqslant 0$ is essential.) – Pedro Dec 06 '13 at 16:37
@PedroTamaroff: this hypothesis would indeed ensure the equivalence, but it was not part of the question. – Siméon Dec 06 '13 at 16:48
Sure. Mine is just a complementary side comment. =) – Pedro Dec 06 '13 at 16:50
How did come up with this example? I'd be glad to know :) – Daniel Gagnon Dec 06 '13 at 17:36
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@DanielGagnon: the statement is true for series of positive numbers, so you have to look for an example of alternating series that is convergent but not absolutely convergent. – Siméon Dec 06 '13 at 18:25

score 4 · Answer 2 · answered Dec 05 '13 at 21:43

4

I suppose you wanted to write that

$1)$

if $\overline \lim(\frac {a_n}{b_n})<+\infty$ and $\sum b_n<+\infty$ then $\sum a_n$ converges too.

$2)$$\underline \lim(\frac {a_n}{b_n})>0$ and $\sum b_n$ diverges then $\sum a_n$ diverges too.

answered Dec 05 '13 at 21:43

Haha

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derivative · Answer 3 · 2013-12-06T16:12:32.600

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If $a_n,b_n\ge 0$ and $\lim_{n\rightarrow\infty} \frac{a_n}{b_n}=1$ then $\exists N_1$ such that,

$\frac{a_n}{b_n}>\frac{1}{2}$ for $n\ge N_1$

which is equivalent to $\quad$$2a_n>b_n$, for $n\ge N_1$

hence, if $\sum_{n}^{\infty} a_n$ converges, then $\sum_{n}^{\infty} 2a_n>\sum_{n}^{\infty} b_n$ also converges.

Similarly $\exists N_2$ such that, $\frac{a_n}{b_n}<\frac{3}{2}$ for $n\ge N_2$

So $a_n<\frac{3b_n}{2}$

If $\sum_{n}^{\infty} b_n$ converges, then also $\sum_{n}^{\infty} a_n$

edited Dec 06 '13 at 16:12

answered Dec 05 '13 at 23:11

derivative

2,450

you seem to be assuming that all the $b_n$'s are positive. The OP did not say anything about the signs of $a_n$ or $b_n$. – Stefan Smith Dec 06 '13 at 02:40
I just noticed that you yourself gave a comment to the question noting a counterexample for sign-changing series: http://math.stackexchange.com/questions/30539/limit-comparison-test-for-alternating-series/30549#30549 – Stefan Smith Dec 06 '13 at 02:41

True/False: $\mathop {\lim }\limits_{n \to \infty } {{{a_n}} \over {{b_n}}} = 1$ implies $\sum {{a_n},\sum {{b_n}} } $ converge or diverge together.

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