Series convergence vs. divergence

[pierce15]

Simple question:

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

 

[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.

 

[henpen]

http://mathoverflow.net/questions/65858/series-whose-convergence-is-not-known has a few examples (some of which understandable, too!). I'm particularly fond of Qiaochu Yuan's).

 

[pierce15]

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.

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