limn→∞anbn=1
Prove the statement implies ∑an,∑bn converge or diverge together.
My guess the statement is true.
if ∑an diverges, then limn→∞an≠0
So,
limn→∞an=L≠0limn→∞anlimn→∞bn=1⇒Llimn→∞bn=1⇒L=limn→∞bn≠0
therefore,
∑bn 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
an=(−1)n√n,andbn=(−1)n√n+1n
The condition an∼bn holds but ∑an is convergent whereas ∑bn is divergent.
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