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P-adic convergence

  1. Jan 25, 2009 #1
    1. The problem statement, all variables and given/known data

    Show, that in [tex]\mathbb{Q}_2[/tex] it holds

    [tex]\sum_{i=1}^{\infty} \frac{2^i}{i} = 0[/tex]

    Hint: Consider [tex]\log_2(-1)[/tex]

    2. Relevant equations

    Just for completeness:

    [tex]\log(1+s) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{s^n}{n}[/tex]

    3. The attempt at a solution

    Actually it's no homework but an exercise from a book. I found already, that

    [tex]\lim_{i \to \infty} |2^i|_2 = 0[/tex]

    which seems to be correct as I found another thread in this forum called "p-adic convergence" which says the same. But that doesn't help me, because

    [tex]\lim_{i \to \infty} \left|\frac{1}{i} \right|_2[/tex]

    does not converge.

    Does anybody know, how to use this hint given in the exercise?

    Thanks in advance.


    Ps.: Hmm, if one looks at [tex]\log(-1)[/tex], then in the upper formula, [tex]s=-2[/tex]. That means [tex]\sum_{i=1}^{\infty} \frac{2^i}{i} = -\log(-1) \overset{?}{=} 0[/tex]. Could that be correct?
    Last edited: Jan 25, 2009
  2. jcsd
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