# Series convergence vs. divergence

1. Mar 10, 2013

### pierce15

Simple question:

Are there any series which we don't know whether or not they converge?

2. Mar 10, 2013

### AlephZero

I'ts fairly easy to invent some. For example, $a_n = 0$ if $2n$ is the sum of two primes, otherwise $a_n = 1$.

First prove the Goldbach conjecture. Checking the convergence or divergence of the series is then trivial

If you don't like that example, think about the consequences of Godel's incompleteness theorems.

3. Mar 10, 2013

### henpen

4. Mar 10, 2013

### pierce15

I don't really understand the implications of the incompleteness theorems, could you briefly explain how they relate to this?