Series for which Convergence is UnknownDate: 11/09/2000 at 11:05:10 From: Veronique Subject: A philosophical question about convergent series I'm a Ph.D. student in commutative algebra working at the Catholic University of Leuven, Belgium. I also have a teaching duty, which is helping the students who have problems with the analysis course. Some time ago, a clever student asked me a question which has puzzled me ever since. It is about series. He asked: 1. Are there series for which nobody knows whether they converge or diverge? 2. Are there series for which it has been proved that you can't prove whether they converge or diverge? This seems to me a more philosophical question, dealing with indecidability. I really don't know the answer. I have already looked in several books, I've asked other mathematicians, I've searched your archives, but no one seems to know the answer. I thank you for your time. With polite regards, Veronique Date: 11/09/2000 at 15:04:23 From: Doctor Rob Subject: Re: A philosophical question about convergent series Thanks for writing to Ask Dr. Math, Veronique. Here is a series whose convergence is unknown. Define a(n) = {1 if n and n+2 are both prime numbers, {0 otherwise. Then SUM a(n) converges if and only if there are only finitely many twin prime pairs. No one knows whether or not there are finitely many twin prime pairs. I have asked a few friends for an example with all terms nonzero. We couldn't think of an example offhand, but we believe that such do exist. One guess I put forward was infinity SUM sin(n)/sqrt(n) n=1 My guess is that this is convergent, but a proof seems very hard. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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