Series for which Convergence is Unknown

Date: 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/