limn→∞anbn=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?
Answer
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.
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