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Series convergence vs. divergence

  1. Mar 10, 2013 #1
    Simple question:

    Are there any series which we don't know whether or not they converge?
     
  2. jcsd
  3. Mar 10, 2013 #2

    AlephZero

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    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 :smile:

    If you don't like that example, think about the consequences of Godel's incompleteness theorems.
     
  4. Mar 10, 2013 #3
  5. Mar 10, 2013 #4
    I don't really understand the implications of the incompleteness theorems, could you briefly explain how they relate to this?
     
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