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### Series for which Convergence is Unknown

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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

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
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/
```
Associated Topics:
High School Sequences, Series

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